Penholodigital squares
Penholodigital squares are perfect square numbers that contain all of the digits from the base in which the number is represented, except for zero, exactly once.
You are encouraged to solve this task according to the task description, using any language you may know.
and holo- (whole, or all)
So, in a particular base, a penholodigital square number will contain all of the digits used in that base (except zero) once, and only once. Base eight penholodigitals contain the digits 1 through 7, base 10, 1 through 9, etc.
- For example
In base 10, 139854276 is a penholodigital square. It is the square of the integer 11826, and contains every digit from 1 through 9 exactly once.
Penholodigital squares can occur in many, though not every, base. They tend to be pretty rare in lower bases.
There is a total of 1 penholodigital squares in base 2: 1² = 1 There is a total of 0 penholodigital squares in base 3: There is a total of 0 penholodigital squares in base 4: There is a total of 0 penholodigital squares in base 5: There is a total of 2 penholodigital squares in base 6: 122² = 15324, 221² = 53241 There is a total of 1 penholodigital squares in base 7: 645² = 623514 There is a total of 1 penholodigital squares in base 8: 2453² = 6532471
- Task
Find and display the total count, and the penholodigital squares and the integers that are squared to produce them, represented in the base in which they are calculated, for bases 9, 10, 11 and 12.
- Stretch
Find and display the total count, and the first and last penholodigital squares and the integers that are squared to produce them, represented in the base in which they are calculated, for bases 13, 14, 15, ... ?
- See also
AppleScript
on penholodigitalSquares(base)
set digits to "123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
set output to {}
set minFromDigits to 1
repeat with d from 2 to (base - 1)
set minFromDigits to minFromDigits * base + d
end repeat
set maxFromDigits to base - 1
repeat with d from (base - 2) to 1 by -1
set maxFromDigits to maxFromDigits * base + d
end repeat
repeat with sqrt from (round (minFromDigits ^ 0.5) rounding up) to (maxFromDigits ^ 0.5 div 1)
set n to sqrt * sqrt
set usedDigitValues to {0}
set OKSoFar to true
repeat (base - 2) times -- until (n < base)
set d to n mod base
if (d is in usedDigitValues) then
set OKSoFar to false
exit repeat
end if
set usedDigitValues's end to d
set n to n div base
end repeat
if ((OKSoFar) and (n is not in usedDigitValues)) then ¬
set end of output to {intToBase(sqrt, base), intToBase(sqrt * sqrt, base)}
end repeat
return output
end penholodigitalSquares
on intToBase(int, base)
if ((int < 0) or (int mod 1 > 0) or (base < 2) or (base > 36)) then return missing value
set digits to "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
set output to digits's character (int mod base + 1)
repeat until (int < base)
set int to int div base
set output to digits's character (int mod base + 1) & output
end repeat
return output
end intToBase
on join(lst, delim)
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to delim
set txt to lst as text
set AppleScript's text item delimiters to astid
return txt
end join
on task()
set output to {}
repeat with base from 9 to 14
set results to penholodigitalSquares(base)
set resultCount to (count results)
if (resultCount > 1) then
set output's end to linefeed & "There are " & resultCount & ¬
(" penholodigital squares in base " & base & ":")
if (base < 13) then
repeat with i from 1 to resultCount by 3
set row to {}
set k to i + 2
if (k > resultCount) then set k to resultCount
repeat with j from i to k
set end of row to join(results's item j, " ^ 2 = ")
end repeat
set output's end to join(row, " ")
end repeat
else
set output's end to "First: " & join(results's beginning, " ^ 2 = ") & ¬
(" Last: " & join(results's end, " ^ 2 = "))
end if
else if (resultCount = 1) then
set output's end to linefeed & "There is 1 penholodigital square in base " & ¬
base & ":"
set output's end to join(results's beginning, " ^ 2 = ")
else
set output's end to linefeed & "There are no penholodigital squares in base " & base
end if
end repeat
return join(output, linefeed)
end task
task()
- Output:
"
There are 10 penholodigital squares in base 9:
3825 ^ 2 = 16328547 3847 ^ 2 = 16523874 4617 ^ 2 = 23875614
4761 ^ 2 = 25487631 6561 ^ 2 = 47865231 6574 ^ 2 = 48162537
6844 ^ 2 = 53184267 7285 ^ 2 = 58624317 7821 ^ 2 = 68573241
8554 ^ 2 = 82314657
There are 30 penholodigital squares in base 10:
11826 ^ 2 = 139854276 12363 ^ 2 = 152843769 12543 ^ 2 = 157326849
14676 ^ 2 = 215384976 15681 ^ 2 = 245893761 15963 ^ 2 = 254817369
18072 ^ 2 = 326597184 19023 ^ 2 = 361874529 19377 ^ 2 = 375468129
19569 ^ 2 = 382945761 19629 ^ 2 = 385297641 20316 ^ 2 = 412739856
22887 ^ 2 = 523814769 23019 ^ 2 = 529874361 23178 ^ 2 = 537219684
23439 ^ 2 = 549386721 24237 ^ 2 = 587432169 24276 ^ 2 = 589324176
24441 ^ 2 = 597362481 24807 ^ 2 = 615387249 25059 ^ 2 = 627953481
25572 ^ 2 = 653927184 25941 ^ 2 = 672935481 26409 ^ 2 = 697435281
26733 ^ 2 = 714653289 27129 ^ 2 = 735982641 27273 ^ 2 = 743816529
29034 ^ 2 = 842973156 29106 ^ 2 = 847159236 30384 ^ 2 = 923187456
There are 20 penholodigital squares in base 11:
42045 ^ 2 = 165742A893 43152 ^ 2 = 173A652894 44926 ^ 2 = 18792A6453
47149 ^ 2 = 1A67395824 47257 ^ 2 = 1A76392485 52071 ^ 2 = 249A758631
54457 ^ 2 = 2719634A85 55979 ^ 2 = 286A795314 59597 ^ 2 = 314672A895
632A4 ^ 2 = 3671A89245 64069 ^ 2 = 376198A254 68335 ^ 2 = 41697528A3
71485 ^ 2 = 46928A7153 81196 ^ 2 = 5A79286413 83608 ^ 2 = 632A741859
86074 ^ 2 = 6713498A25 89468 ^ 2 = 7148563A29 91429 ^ 2 = 76315982A4
93319 ^ 2 = 795186A234 A3A39 ^ 2 = 983251A764
There are 23 penholodigital squares in base 12:
117789 ^ 2 = 135B7482A69 16357B ^ 2 = 23A5B976481 16762B ^ 2 = 24AB5379861
16906B ^ 2 = 25386749BA1 173434 ^ 2 = 26B859A3714 178278 ^ 2 = 2835BA17694
1A1993 ^ 2 = 34A8125B769 1A3595 ^ 2 = 354A279B681 1B0451 ^ 2 = 3824B7569A1
1B7545 ^ 2 = 3A5B2487961 2084A9 ^ 2 = 42A1583B769 235273 ^ 2 = 5287BA13469
2528B5 ^ 2 = 5B23A879641 25B564 ^ 2 = 62937B5A814 262174 ^ 2 = 63A8527B194
285A44 ^ 2 = 73B615A8294 29A977 ^ 2 = 7B9284A5361 2A7617 ^ 2 = 83AB5479261
2B0144 ^ 2 = 8617B35A294 307381 ^ 2 = 93825A67B41 310828 ^ 2 = 96528AB7314
319488 ^ 2 = 9AB65823714 319A37 ^ 2 = 9B2573468A1
There are no penholodigital squares in base 13
There are 160 penholodigital squares in base 14:
First: 1129535 ^ 2 = 126A84D79C53B Last: 3A03226 ^ 2 = DB3962A7541C8"
C++
#include <algorithm>
#include <cmath>
#include <cstdint>
#include <iostream>
#include <set>
#include <string>
#include <vector>
const std::string digits = "0123456789abcdef";
std::string convert_decimal_to_base(uint64_t number, const uint32_t& radix) {
std::string result = "";
while ( number > 0 ) {
result += digits[number % radix];
number /= radix;
}
std::reverse(result.begin(), result.end());
return result;
}
int main() {
const std::vector<uint32_t> largest_prime_factors = { 1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5 };
for ( size_t radix = 2; radix <= 16; ++radix ) {
std::vector<std::string> penholo = { };
std::vector<std::string> penholo_squares = { };
std::string radix_digits = digits.substr(1, radix - 1);
uint32_t min = std::ceil(std::sqrt(std::stoll(radix_digits, nullptr, radix)));
std::reverse(radix_digits.begin(), radix_digits.end());
const uint32_t max = std::floor(std::sqrt(std::stoll(radix_digits, nullptr, radix)));
const uint32_t divisor = largest_prime_factors[radix - 2];
min += ( min % divisor == 0 ) ? 0 : ( divisor - min % divisor );
for ( uint64_t number = min; number <= max; number += divisor ) {
std::string square = convert_decimal_to_base(number * number, radix);
square.erase(std::remove(square.begin(), square.end(), '0'), square.end());
if ( std::set<char>{ square.begin(), square.end() }.size() == radix - 1 ) {
penholo.emplace_back(convert_decimal_to_base(number, radix));
penholo_squares.emplace_back(square);
}
}
std::cout << "There is a total of " << penholo.size()
<< " penholodigital squares in base " << radix << ":" << "\n";
if ( radix <= 13 ) {
for ( uint64_t i = 0; i < penholo.size(); ++i ) {
std::cout << penholo[i] + "² = " + penholo_squares[i]
+ ( i % 3 == 2 ? "\n" : " " );
}
std::cout << ( penholo.size() % 3 == 0 ? "\n" : "\n\n" );
} else {
std::cout << penholo.front() << "² = " << penholo_squares.front() << " ... "
<< penholo.back() << "² = " << penholo_squares.back() << "\n\n";
}
}
}
- Output:
There is a total of 1 penholodigital squares in base 2: 1² = 1 There is a total of 0 penholodigital squares in base 3: There is a total of 0 penholodigital squares in base 4: There is a total of 0 penholodigital squares in base 5: There is a total of 2 penholodigital squares in base 6: 122² = 15324 221² = 53241 There is a total of 1 penholodigital squares in base 7: 645² = 623514 There is a total of 1 penholodigital squares in base 8: 2453² = 6532471 There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There is a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126a84d79c53b ... 3a03226² = db3962a7541c8 There is a total of 419 penholodigital squares in base 15: 4240c58² = 12378da5b6ec94 ... ee25e4a² = ed4c93285671ba There is a total of 740 penholodigital squares in base 16: 11156eb6² = 123da7f85bce964 ... 3fd8f786² = fec81b69573da24
Delphi
{Library routine included here for clarity}
function GetRadixString(L: int64; Radix: Byte): string;
{Converts integer a string of any radix}
const RadixChars: array[0..35] Of char =
('0', '1', '2', '3', '4', '5', '6', '7',
'8', '9', 'A', 'B', 'C', 'D', 'E', 'F',
'G','H', 'I', 'J', 'K', 'L', 'M', 'N',
'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V',
'W', 'X', 'Y', 'Z');
var I: cardinal;
var S: string;
var Sign: string[1];
begin
Result:='';
If (L < 0) then
begin
Sign:='-';
L:=Abs(L);
end
else Sign:='';
S:='';
repeat
begin
I:=L mod Radix;
S:=RadixChars[I] + S;
L:=L div Radix;
end
until L = 0;
Result:=Sign + S;
end;
function IsPenholoSquare(N: int64; Radix: byte): boolean;
{Test if N is a Penholodigital square}
var S: string;
var SL: TStringList;
var I: integer;
begin
Result:=False;
SL:=TStringList.Create;
try
{Get text version of number in the radix}
S:=GetRadixString(N,Radix);
{Reject any number with a zero in it}
if Pos('0',S)>0 then exit;
{Make sure string list doesn't duplicates}
SL.Sorted:=True;
SL.Duplicates:=dupIgnore;
{Insert each digit as one string in the list}
for I:=1 to Length(S) do SL.Add(S[I]);
{Same number of unique digits as radix-1? }
Result:=SL.Count=Radix-1;
finally SL.Free; end;
end;
function GetNDigitNumber(Digits,Radix: integer): extended;
{Get smallest N-digit number of specified radix}
begin
Result:=Power(10,(Digits-1) * Log(Radix));
end;
procedure GetStartStop(Radix: integer; var Start,Stop: int64);
{Start/Stop = Range of numbers to check}
{Since zero is not allowed, we want number width digits = Radix-1}
{Start/Stop squared = Smallest largest number that could match}
var S1,S2,Sqrt1,Sqrt2: extended;
begin
Start:=Trunc(Sqrt(GetNDigitNumber(Radix-1,Radix)));
Stop:=Trunc(Sqrt(GetNDigitNumber(Radix,Radix)-1));
end;
procedure FindAllPenholoSquares(Memo: TMemo; Radix: integer);
{Find all the Penholodigital squares for the specified radix}
var I,Start,Stop: int64;
var S,H: string;
var Cnt: integer;
begin
Cnt:=0;
{Get the range to look over}
GetStartStop(Radix,Start,Stop);
{Scan all the number for Penholodigital squares}
I:=Start;
while I<=Stop do
begin
if IsPenholoSquare(I*I,Radix) then
begin
Inc(Cnt);
S:=S+GetRadixString(I,Radix)+'² = '+GetRadixString(I*I,Radix);
if I<Stop then S:=S+' ';
If (Cnt mod 3)=0 then S:=S+CRLF;
end;
Inc(I);
end;
H:=CRLF+'Penholodigital squares in base '+IntToStr(Radix)+CRLF;
H:=H+'Start: '+GetRadixString(Start,Radix)+'² = '+GetRadixString(Start*Start,Radix)+CRLF;
H:=H+'Stop: '+GetRadixString(Stop,Radix)+'² = '+GetRadixString(Stop*Stop,Radix)+CRLF;
H:=H+'Count='+IntToStr(Cnt)+CRLF;
Memo.Lines.Add(H+S);
end;
procedure ShowPenholoSquares(Memo: TMemo);
{Show Penholodigital squares for various radices}
var N,C: int64;
begin
FindAllPenholoSquares(Memo,9);
FindAllPenholoSquares(Memo,10);
FindAllPenholoSquares(Memo,11);
FindAllPenholoSquares(Memo,12);
FindAllPenholoSquares(Memo,13);
FindAllPenholoSquares(Memo,14);
end;
- Output:
Penholodigital squares in base 9 Start: 3000² = 10000000 Stop: 8888² = 88870001 Count=10 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 Penholodigital squares in base 10 Start: 10000² = 100000000 Stop: 31622² = 999950884 Count=30 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 Penholodigital squares in base 11 Start: 33534² = AAAA63735 Stop: AAAAA² = AAAA900001 Count=20 42045² = 165742A893 43152² = 173A652894 44926² = 18792A6453 47149² = 1A67395824 47257² = 1A76392485 52071² = 249A758631 54457² = 2719634A85 55979² = 286A795314 59597² = 314672A895 632A4² = 3671A89245 64069² = 376198A254 68335² = 41697528A3 71485² = 46928A7153 81196² = 5A79286413 83608² = 632A741859 86074² = 6713498A25 89468² = 7148563A29 91429² = 76315982A4 93319² = 795186A234 A3A39² = 983251A764 Penholodigital squares in base 12 Start: 100000² = 10000000000 Stop: 3569B7² = BBBBB983821 Count=23 117789² = 135B7482A69 16357B² = 23A5B976481 16762B² = 24AB5379861 16906B² = 25386749BA1 173434² = 26B859A3714 178278² = 2835BA17694 1A1993² = 34A8125B769 1A3595² = 354A279B681 1B0451² = 3824B7569A1 1B7545² = 3A5B2487961 2084A9² = 42A1583B769 235273² = 5287BA13469 2528B5² = 5B23A879641 25B564² = 62937B5A814 262174² = 63A8527B194 285A44² = 73B615A8294 29A977² = 7B9284A5361 2A7617² = 83AB5479261 2B0144² = 8617B35A294 307381² = 93825A67B41 310828² = 96528AB7314 319488² = 9AB65823714 319A37² = 9B2573468A1 Penholodigital squares in base 13 Start: 37B451² = CCCCC61C7A1 Stop: CCCCCC² = CCCCCB000001 Count=0 Penholodigital squares in base 14 Start: 1000000² = 1000000000000 Stop: 3A55171² = DDDDDD67CDA01 Count=160 1129535² = 126A84D79C53B 1145393² = 12A68DB5374C9 117D854² = 134596CAB87D2 11865A8² = 1356BC247D9A8 11888A3² = 135BAD2678C49 1213ACA² = 146CA78D53B92 1225AA9² = 149785AC62D3B 1235DC3² = 14BC765DA8329 1279052² = 157AB3D289C64 12894A5² = 15A29D468C73B 128B953² = 15A8364DC7B29 12B30AC² = 1623D89B7A5C4 12D71B4² = 167B48ACD3952 13485D5² = 1763D8CA9245B 136C494² = 17BD495AC8632 138BC89² = 182C6D7534A9B 1393B93² = 183D645B72AC9 13A0084² = 185B36D7A4C92 1461C78² = 1A3D7C6925B48 1470999² = 1A6548739C2DB 149AA14² = 1AD8546C37B92 1508CA3² = 1BCA2D5736849 151DB35² = 1C25A647D893B 152649C² = 1C398AD765B24 1576CAB² = 1D38AB65C7249 157C257² = 1D4965C8BA237 15809D3² = 1D527B3A6C489 1586757² = 1D64BA5C89237 16D1B9B² = 23457BCD861A9 180D3D1² = 26ADB397548C1 183B736² = 2761354D9CBA8 1877349² = 283CAD946157B 190AA09² = 29D738461AC5B 193AAA9² = 2A9657C148D3B 19910D6² = 2BD357619AC48 19A4195² = 2C35164A98D7B 1A1030B² = 2D56B71C34A89 1A18B62² = 2D785C3691BA4 1B11831² = 32CDA84759B61 1B84906² = 34B17956CDA28 1B909D9² = 34DA578296C1B 1BBB53A² = 359C84AD761B2 1C0C851² = 3674BC2598DA1 1C21091² = 36BA29D587C41 1C25991² = 36CD78A95B241 1C9891C² = 38C2A5DB71964 1CA7D36² = 391B6C2475DA8 1CC1274² = 3978B56A4CD12 1D034D5² = 3A1CD2476985B 1D18196² = 3A716D45B92C8 1D53476² = 3B74AC96D1528 1D82B89² = 3C4DA1285769B 1D891AA² = 3C6A79415D8B2 1D9C33B² = 3CB8216D57A49 1DD841C² = 3DC529A78B164 205B2B8² = 41952D3A67BC8 206B1C7² = 41D5BA69C8237 208A775² = 427589D63AC1B 20D0164² = 43AC6DB981572 216BBBA² = 46185C9ADB732 217972B² = 46518B2C3D7A9 21842D9² = 467A53289DC1B 21B1038² = 4756D93B12CA8 21BD039² = 47921586CAD3B 22188B7² = 48AD25C19B637 2220ADA² = 48CB93D567A12 22228DA² = 48D593BA76C12 2257406² = 49DB5A71C3628 226D835² = 4A619C8D5273B 227211B² = 4A6C7D1385B29 22835B8² = 4AB96D12753C8 22CB813² = 4C3B528DA1769 2304629² = 4CA59718362DB 2402131² = 5329784CDAB61 24080CD² = 5348C7A6B9D21 24141A4² = 537CD86A941B2 244C9D8² = 54A7B93CD1628 249A12B² = 564CBD78321A9 24DD54B² = 57AB12834D6C9 255AD7A² = 59A6148D3CB72 255C168² = 59AB746D1C328 256AD5B² = 5A12D34C78B69 2576D8B² = 5A48C673B1D29 2586549² = 5A9328C641D7B 2588178² = 5A9B6C21D7348 259CCB1² = 5B23CA79D4681 25C895B² = 5C1438ADB2769 25CB949² = 5C24A986D317B 25D251B² = 5C3D784AB6129 261D43A² = 5D36C7418A9B2 26280BA² = 5D6789AC41B32 262BA53² = 5D7B861CA4329 2634B3A² = 5DA41C73689B2 267941A² = 61374A895BCD2 270A8B1² = 63AB275D49C81 276CC76² = 65DBC7A914328 27A7D0D² = 67458DB39A2C1 27D30D1² = 683DB5A9742C1 2810A05² = 68D5A943C721B 2811842² = 68DA3C95B1724 2837D42² = 69C5B1738AD24 2859D11² = 6A975BC843D21 296B7C5² = 7254638A1DC9B 29D7768² = 74D6CA935B128 2AA5C1C² = 793185C2DAB64 2AD13CB² = 7A3D28B14C569 2B20281² = 7B63459CA8D21 2B21C1A² = 7B6CA139854D2 2B5A308² = 7CD13AB529648 2C40134² = 83CD916A574B2 2CB5679² = 86DA721493C5B 2D26547² = 8915B2ADC4637 2D740A3² = 8B1D75A2C3649 2DA0095² = 8C415DA92637B 30BB095² = 95162DCA8437B 30C0C0C² = 95317BC2D68A4 30D632C² = 95B83DC167A24 30D6D04² = 95BC8347AD612 3150917² = 9841B5A6CD237 3191C1C² = 9A175D382CB64 3193C92² = 9A261B73CD584 31C0878² = 9B5A1726CD348 3206366² = 9C6B3D2A51748 3276C18² = A195234DB7C68 32781B7² = A19D2CB854637 3307D3A² = A4C918563D7B2 33867CD² = A87B49365CD21 33A00DA² = A93B857C4D612 33A3BD9² = A95847C2D361B 340A621² = AB6C8D5793241 3441C78² = AD139726C5B48 34B4836² = B27146DC359A8 35A38CB² = B8A5D174C2369 35B0232² = B917A2856D3C4 35D5841² = BA3C52D796481 35DDC8B² = BA7D48361C529 363A5CD² = BC5A68D479321 369A308² = C152B73DA9648 36C9531² = C2B9A43D87561 36D9588² = C349B5721AD68 373627B² = C5321A47BD689 3736C8D² = C53729468BDA1 37574A3² = C63B78A1D5249 3759087² = C64981ABD5237 37861A4² = C7A5D493861B2 37CB72A² = C9D84AB316752 381827A² = CB548A369D172 38534A6² = CD37B496512A8 3859842² = CD6AB81395724 38AA3D5² = D21396874AC5B 38BA9A1² = D28A3B549C761 3900B4C² = D3B5C9612A784 390A926² = D42795B36A1C8 39407D3² = D5C742BA31689 394C2D9² = D64895A723C1B 397025D² = D764AB895C321 39A1835² = D912C45A6873B 39B0934² = D981746A53CB2 39C021C² = DA135B28C7964 3A03226² = DB3962A7541C8 Elapsed Time: 03:03.390 min
FreeBASIC
#define floor(x) ((x*2.0-0.5) Shr 1)
Dim Shared As Double bbase ' Number bbase being used [9..14]
Sub NumOut(n As Double) ' Display n in the specified bbase
Dim As Integer remain = Fix(n Mod bbase)
n = Floor(n / bbase)
If n <> 0 Then NumOut(n)
Print Chr(remain + Iif(remain <= 9, Asc("0"), Asc("A")-10));
End Sub
Sub ShowPenSq(n As Double) ' Display n = n^2
NumOut(n)
'Print Chr(253); " = ";
Print "^2 = ";
NumOut(n * n)
Print " ";
End Sub
Function isPenholodigital(n As Double) As Boolean
Dim As Integer used, remain
used = 1 ' can't contain 0
While n <> 0
remain = Fix(n Mod bbase)
n = Floor(n / bbase)
If (used And (1 Shl remain)) Then Return False
used = used Or (1 Shl remain)
Wend
Return (used = (1 Shl Fix(bbase)) - 1)
End Function
Dim As Uinteger cnt
Dim As Double n, limite, n1, n2
bbase = 9
Do
cnt = 0
n1 = 0
n = Floor(Sqr(bbase ^ (bbase-2)))
limite = Sqr(bbase ^ (bbase-1))
Do
If isPenholodigital(n * n) Then
cnt += 1
If n1 = 0 Then n1 = n
If bbase <= 12 Then
ShowPenSq(n)
If (cnt Mod 3) = 0 Then Print
Else
n2 = n
End If
End If
n += 1
Loop Until n >= limite
If (cnt Mod 3) <> 0 Then Print
Print "There are "; cnt; " penholodigital squares in base "; Fix(bbase)
If bbase >= 13 And cnt > 0 Then
ShowPenSq(n1)
Print ".. ";
ShowPenSq(n2)
End If
Print
bbase += 1
Loop Until bbase >= 15
Sleep
- Output:
Similar as XPLo entry.
Go
package main
import (
"fmt"
"math"
"rcu"
"strconv"
)
func reverse(s string) string {
r := make([]byte, len(s))
for i := 0; i < len(s); i++ {
r[i] = s[len(s)-1-i]
}
return string(r)
}
func main() {
primes := []int{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}
digits := "123456789ABCDEF"
for b := 9; b <= 16; b++ {
master := 1
for d := 1; d < b; d++ {
master *= primes[d-1]
}
var phd []int
smin, _ := strconv.ParseInt(digits[0:b-1], b, 64)
min := int(math.Ceil(math.Sqrt(float64(smin))))
smax, _ := strconv.ParseInt(reverse(digits[0:b-1]), b, 64)
max := int(math.Floor(math.Sqrt(float64(smax))))
factors := rcu.PrimeFactors(b - 1)
div := factors[len(factors)-1]
for i := min; i <= max; i++ {
if (i % div) != 0 {
continue
}
sq := i * i
digs := rcu.Digits(sq, b)
containsZero := false
key := 1
for _, dig := range digs {
if dig == 0 {
containsZero = true
break
}
key *= primes[dig-1]
}
if containsZero {
continue
}
if key == master {
phd = append(phd, i)
}
}
fmt.Println("There is a total of", len(phd), "penholodigital squares in base", b, "\b:")
if b > 13 {
phd = []int{phd[0], phd[len(phd)-1]}
}
for i := 0; i < len(phd); i++ {
sq2 := phd[i] * phd[i]
fmt.Printf("%s² = %s ", strconv.FormatInt(int64(phd[i]), b), strconv.FormatInt(int64(sq2), b))
if (i+1)%3 == 0 {
fmt.Println()
}
}
if len(phd)%3 != 0 {
fmt.Println()
}
fmt.Println()
}
}
- Output:
There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There is a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126a84d79c53b 3a03226² = db3962a7541c8 There is a total of 419 penholodigital squares in base 15: 4240c58² = 12378da5b6ec94 ee25e4a² = ed4c93285671ba There is a total of 740 penholodigital squares in base 16: 11156eb6² = 123da7f85bce964 3fd8f786² = fec81b69573da24
J
Implementation:
digch=: a.{~;48 97(+i.)&.>10 26
brep=: (digch {~ #.inv)&.>
penholod=: {{
F=: >.%:y#.D=:}.i.y
C=: <.%:y#.}:i.-y
ok=: (D */@e. y #.inv ])"0
(#~ok) *:F+i.1+C-F
}}
task=: {{
sq=. penholod y
hd=. ,:(#sq),&":' penholodigital squares in base ',":y
hd,(*#sq)#names (y brep sq),each '=',each(y brep %:sq),each<'²'
}}
stretch=: {{
sq=. penholod y
hd=. ,:(#sq),&":' penholodigital squares in base ',":y
hd,(*#sq)#names ({.,'...';{:) (y brep sq),each '=',each(y brep %:sq),each<'²'
}}
Task examples:
task 9
10 penholodigital squares in base 9
16328547=3825² 16523874=3847² 23875614=4617² 25487631=4761² 47865231=6561² 48162537=6574²
53184267=6844² 58624317=7285² 68573241=7821² 82314657=8554²
task 10
30 penholodigital squares in base 10
139854276=11826² 152843769=12363² 157326849=12543² 215384976=14676² 245893761=15681² 254817369=15963²
326597184=18072² 361874529=19023² 375468129=19377² 382945761=19569² 385297641=19629² 412739856=20316²
523814769=22887² 529874361=23019² 537219684=23178² 549386721=23439² 587432169=24237² 589324176=24276²
597362481=24441² 615387249=24807² 627953481=25059² 653927184=25572² 672935481=25941² 697435281=26409²
714653289=26733² 735982641=27129² 743816529=27273² 842973156=29034² 847159236=29106² 923187456=30384²
task 11
20 penholodigital squares in base 11
165742a893=42045² 173a652894=43152² 18792a6453=44926² 1a67395824=47149² 1a76392485=47257²
249a758631=52071² 2719634a85=54457² 286a795314=55979² 314672a895=59597² 3671a89245=632a4²
376198a254=64069² 41697528a3=68335² 46928a7153=71485² 5a79286413=81196² 632a741859=83608²
6713498a25=86074² 7148563a29=89468² 76315982a4=91429² 795186a234=93319² 983251a764=a3a39²
task 12
23 penholodigital squares in base 12
135b7482a69=117789² 23a5b976481=16357b² 24ab5379861=16762b² 25386749ba1=16906b² 26b859a3714=173434²
2835ba17694=178278² 34a8125b769=1a1993² 354a279b681=1a3595² 3824b7569a1=1b0451² 3a5b2487961=1b7545²
42a1583b769=2084a9² 5287ba13469=235273² 5b23a879641=2528b5² 62937b5a814=25b564² 63a8527b194=262174²
73b615a8294=285a44² 7b9284a5361=29a977² 83ab5479261=2a7617² 8617b35a294=2b0144² 93825a67b41=307381²
96528ab7314=310828² 9ab65823714=319488² 9b2573468a1=319a37²
stretch 13
0 penholodigital squares in base 13
stretch 14
160 penholodigital squares in base 14
126a84d79c53b=1129535² ...
db3962a7541c8=3a03226²
stretch 15
419 penholodigital squares in base 15
12378da5b6ec94=4240c58² ...
ed4c93285671ba=ee25e4a²
stretch 16
740 penholodigital squares in base 16
123da7f85bce964=11156eb6² ...
fec81b69573da24=3fd8f786²
NB. this is getting to be obnoxiously long in terms of time...
Java
import java.util.ArrayList;
import java.util.List;
public final class PenholodigitalSquares {
public static void main(String[] args) {
String digits = "0123456789abcdef";
List<Integer> largestPrimeFactors = List.of( 1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5 );
for ( int radix = 2; radix <= 16; radix++ ) {
List<String> penholo = new ArrayList<String>();
List<String> penholoSquares = new ArrayList<String>();
String radixDigits = digits.substring(1, radix);
String reversed = new StringBuilder(radixDigits).reverse().toString();
int min = (int) Math.ceil(Math.sqrt(Long.parseLong(radixDigits, radix)));
final int max = (int) Math.floor(Math.sqrt(Long.parseLong(reversed, radix)));
final int divisor = largestPrimeFactors.get(radix - 2);
min += ( min % divisor == 0 ) ? 0 : ( divisor - min % divisor );
for ( int number = min; number <= max; number += divisor ) {
String square = Long.toString((long) number * number, radix);
if ( square.chars().filter( i -> i != 48 ).distinct().count() == radix - 1 ) {
penholo.addLast(Long.toString(number, radix));
penholoSquares.addLast(square);
}
}
System.out.println(
"There is a total of " + penholo.size() + " penholodigital squares in base " + radix + ":");
if ( radix <= 13 ) {
for ( int i = 0; i < penholo.size(); i++ ) {
System.out.print(penholo.get(i) + "² = " + penholoSquares.get(i)
+ ( i % 3 == 2 ? "\n" : " " ));
}
System.out.println( penholo.size() % 3 == 0 ? "" : System.lineSeparator());
} else {
System.out.println(penholo.getFirst() + "² = " + penholoSquares.getFirst() + " ... "
+ penholo.getLast() + "² = " + penholoSquares.getLast() + System.lineSeparator());
}
}
}
}
- Output:
There is a total of 1 penholodigital squares in base 2: 1² = 1 There is a total of 0 penholodigital squares in base 3: There is a total of 0 penholodigital squares in base 4: There is a total of 0 penholodigital squares in base 5: There is a total of 2 penholodigital squares in base 6: 122² = 15324 221² = 53241 There is a total of 1 penholodigital squares in base 7: 645² = 623514 There is a total of 1 penholodigital squares in base 8: 2453² = 6532471 There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There is a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126a84d79c53b ... 3a03226² = db3962a7541c8 There is a total of 419 penholodigital squares in base 15: 4240c58² = 12378da5b6ec94 ... ee25e4a² = ed4c93285671ba There is a total of 740 penholodigital squares in base 16: 11156eb6² = 123da7f85bce964 ... 3fd8f786² = fec81b69573da24
jq
Adapted from Python
Also works with gojq, the Go implementation of jq, and with fq
Note that the definition of `_nwise/1` may be omitted if using jq.
General Utilities
# This def may be omitted if using jq
def _nwise($n):
def nw: if length <= $n then . else .[0:$n] , (.[$n:] | nw) end;
nw;
# Evaluate SIGMA $x^k * $p[k] for k=0...
def evalpoly($x; $p):
reduce range(0;p|length) as $i ({power: 1, sum:0};
.sum += .power * $p[$i]
| .power *= $x)
| .sum;
# Convert the input integer to a string in the specified base (2 to 36 inclusive)
def convert(base):
def stream:
recurse(if . >= base then ./base|floor else empty end) | . % base ;
[stream] | reverse
| if base < 10 then map(tostring) | join("")
elif base <= 36 then map(if . < 10 then 48 + . else . + 87 end) | implode
else error("base too large")
end
end;
# If $j is 0, then an error condition is raised;
# otherwise, assuming infinite-precision integer arithmetic,
# if the input and $j are integers, then the result will be an integer.
def idivide($j):
. as $i
| ($i % $j) as $mod
| ($i - $mod) / $j ;
# input should be a non-negative integer for accuracy
# but may be any non-negative finite number
def isqrt:
def irt:
. as $x
| 1 | until(. > $x; . * 4) as $q
| {$q, $x, r: 0}
| until( .q <= 1;
.q |= idivide(4)
| .t = .x - .r - .q
| .r |= idivide(2)
| if .t >= 0
then .x = .t
| .r += .q
else .
end)
| .r ;
if type == "number" and (isinfinite|not) and (isnan|not) and . >= 0
then irt
else "isqrt requires a non-negative integer for accuracy" | error
end ;
# Input: an integer base 10
# Output: an array of the digits (characters) if . were printed in base $base
def digits($base):
convert($base) | tostring | [explode[] | [.] | implode];
The Task
# emit an array of [$n,$sq] values where $n is a penholodigital square in the given base
# and $n and $sq are integers expressed in that base
def penholodigital($base):
{ hi: (evalpoly($base; [range(1;$base)])|isqrt),
lo: (evalpoly($base; [range(base-1; 0; -1)]) | isqrt) # evalpoly(base, base-1:-1:1)
}
| reduce range(.lo; .hi+1) as $n (null;
($n * $n) as $sq
| ($sq | digits($base)) as $digits
| if "0" | IN($digits[]) then .
elif (($digits | length) == $base - 1) and (($digits | unique | length) == $base-1)
then . + [[($n | convert(base)), ($sq | convert(base))]]
else .
end );
def task(a;b):
range(a;b) as $base
| penholodigital($base)
| "\n\nThere are \(length) penholodigital squares in base \($base):",
(_nwise(3)
| map("\(.[0])² = \(.[1])" )
| join(" "));
task(9;13)
- Output:
There are 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There are 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There are 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There are 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1
Julia
""" rosettacode.org task Penholodigital_squares """
function penholodigital(base)
penholodigitals = Int[]
hi, lo = isqrt(evalpoly(base, 1:base-1)), isqrt(evalpoly(base, base-1:-1:1))
for n in lo:hi
dig = digits(n * n; base)
0 in dig && continue
if all(i -> count(==(i), dig) == 1, 1:base-1)
push!(penholodigitals, n * n)
end
end
return penholodigitals
end
for j in 9:16
allpen = penholodigital(j)
println("\n\nThere is a total of $(length(allpen)) penholodigital squares in base $j:")
for (i, n) in (j < 14 ? enumerate(allpen) : enumerate([allpen[begin], allpen[end]]))
print(string(isqrt(n), base=j), "² = ", string(n, base=j), i %3 == 0 ? "\n" : " ")
end
end
- Output:
There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There is a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126a84d79c53b 3a03226² = db3962a7541c8 There is a total of 419 penholodigital squares in base 15: 4240c58² = 12378da5b6ec94 ee25e4a² = ed4c93285671ba There is a total of 740 penholodigital squares in base 16: 11156eb6² = 123da7f85bce964 3fd8f786² = fec81b69573da24
Extended version
Theoretically, the program should be able to handle bases up to 30, but in practice that would take it far too long.
function penholodigital(base)
penholodigitals = [typeof(base)[] for _ in 1:Threads.nthreads()]
digitbuf = [zeros(typeof(base), base-1) for _ in 1:Threads.nthreads()]
hi, lo = isqrt(evalpoly(base, 1:base-1)), isqrt(evalpoly(base, base-1:-1:1))
@Threads.threads for n in lo:hi
dig = digitbuf[Threads.threadid()]
digits!(dig, n * n; base)
0 in dig && continue
if all(i -> count(==(i), dig) == 1, 1:base-1)
push!(penholodigitals[Threads.threadid()], n * n)
end
end
return sort!(vcat(penholodigitals...))
end
for j in 9:19
@time begin
allpen = penholodigital(j < 17 ? j : Int128(j))
println("There are a total of $(length(allpen)) penholodigital squares in base $j:")
if length(allpen) > 0
for (i, n) in (j < 14 ? enumerate(allpen) : enumerate([allpen[begin], allpen[end]]))
print(string(isqrt(n), base=j), "² = ", string(n, base=j), i %3 == 0 ? "\n" : " ")
end
end
end
println("\n")
end
- Output:
There are a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 0.180057 seconds (213.67 k allocations: 10.895 MiB, 98.16% compilation time) There are a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 0.008105 seconds (770 allocations: 28.117 KiB) There are a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 0.008614 seconds (538 allocations: 21.164 KiB) There are a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1 0.021154 seconds (606 allocations: 23.141 KiB) There are a total of 0 penholodigital squares in base 13: 0.093971 seconds (67 allocations: 6.148 KiB) There are a total of 160 penholodigital squares in base 14: 1129535² = 126a84d79c53b 3a03226² = db3962a7541c8 0.678507 seconds (148 allocations: 12.523 KiB, 0.55% compilation time) There are a total of 419 penholodigital squares in base 15: 4240c58² = 12378da5b6ec94 ee25e4a² = ed4c93285671ba 4.194926 seconds (137 allocations: 22.289 KiB) There are a total of 740 penholodigital squares in base 16: 11156eb6² = 123da7f85bce964 3fd8f786² = fec81b69573da24 10.843582 seconds (137 allocations: 24.789 KiB) There are a total of 0 penholodigital squares in base 17: 345.116100 seconds (335.36 k allocations: 17.403 MiB, 0.08% compilation time) There are a total of 5116 penholodigital squares in base 18: 11150fc0g² = 123cd8abh5g79f6e4 44422dd18² = hgef25738d496bc1a 2584.183274 seconds (332.23 k allocations: 17.565 MiB, 0.00% gc time, 0.01% compilation time) There are a total of 47677 penholodigital squares in base 19: 4b802235a² = 1234978cibd6gfhea5 ii844bia2² = ihg8f71c6da59be324 20734.916185 seconds (169 allocations: 2.163 MiB)
Nim
Translation with several modifications to make things easier in Nim.
import std/[algorithm, math, strformat, sugar]
const
Primes = [1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
Digits = "0123456789ABCDEF"
func digits(n, base: Positive): seq[int] =
## Return the digits of "n" in given base (least significant
## digits first).
var n = n.Natural
while n != 0:
result.add n mod base
n = n div base
func parseInt(d: openArray[char]; base: Positive): int =
## Convert a sequence of characters in given base to an integer.
for c in d:
result = result * base + (if c <= '9': ord(c) - ord('0') else: ord(c) - ord('A') + 10)
func toBase(n: Natural; base: Positive): string =
## Return the string representation of "n" in given base.
let d = n.digits(base)
for i in countdown(d.high, 0):
result.add Digits[d[i]]
func lastPrimeFactor(n: Positive): int =
## Return the last prime factor of "n".
var n = n
if (n and 1) == 0:
result = 2
while true:
n = n shr 1
if (n and 1) != 0: break
var d = 3
while d * d <= n:
if n mod d == 0:
result = d
while true:
n = n div d
if n mod d != 0: break
inc d, 2
if n > result: result = n
for b in 9..16:
let master = Primes[1..<b]
var phd: seq[int]
let digits = Digits[1..<b]
let min = sqrt(digits.parseInt(b).toFloat).ceil.toInt
let max = sqrt(digits.reversed.parseInt(b).toFloat).ceil.toInt
let divider = lastPrimeFactor(b - 1)
# Build the list of square roots of pendholodigital squares.
for i in min..max:
if i mod divider != 0: continue
let sq = i * i
let digs = sq.digits(b)
if 0 in digs: continue
var key = collect(for dig in digs: Primes[dig])
if sorted(key) == master: phd.add i
echo &"There is a total of {phd.len} penholodigital squares in base {b}:"
if b > 13: phd = @[phd[0], phd[^1]]
for i in 0..phd.high:
stdout.write &" {phd[i].toBase(b)}² = {(phd[i]^2).toBase(b)}"
if i mod 3 == 2: echo()
if phd.len mod 3 != 0: echo()
echo()
- Output:
There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742A893 43152² = 173A652894 44926² = 18792A6453 47149² = 1A67395824 47257² = 1A76392485 52071² = 249A758631 54457² = 2719634A85 55979² = 286A795314 59597² = 314672A895 632A4² = 3671A89245 64069² = 376198A254 68335² = 41697528A3 71485² = 46928A7153 81196² = 5A79286413 83608² = 632A741859 86074² = 6713498A25 89468² = 7148563A29 91429² = 76315982A4 93319² = 795186A234 A3A39² = 983251A764 There is a total of 23 penholodigital squares in base 12: 117789² = 135B7482A69 16357B² = 23A5B976481 16762B² = 24AB5379861 16906B² = 25386749BA1 173434² = 26B859A3714 178278² = 2835BA17694 1A1993² = 34A8125B769 1A3595² = 354A279B681 1B0451² = 3824B7569A1 1B7545² = 3A5B2487961 2084A9² = 42A1583B769 235273² = 5287BA13469 2528B5² = 5B23A879641 25B564² = 62937B5A814 262174² = 63A8527B194 285A44² = 73B615A8294 29A977² = 7B9284A5361 2A7617² = 83AB5479261 2B0144² = 8617B35A294 307381² = 93825A67B41 310828² = 96528AB7314 319488² = 9AB65823714 319A37² = 9B2573468A1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126A84D79C53B 3A03226² = DB3962A7541C8 There is a total of 419 penholodigital squares in base 15: 4240C58² = 12378DA5B6EC94 EE25E4A² = ED4C93285671BA There is a total of 740 penholodigital squares in base 16: 11156EB6² = 123DA7F85BCE964 3FD8F786² = FEC81B69573DA24
Pascal
Free Pascal
Nearly copy and paste of pandigital square numbers.
Now using the right step size and startvalue
I think base 16 is the limit. 1234...FG is to big for Uint64. GMP/ MPinteger is required.
program penholodigital;
//Find the smallest number n to base b, so that n*n includes all
//digits of base b without 0
{$IFDEF FPC}{$MODE DELPHI}{$Optimization ON,All}{$ENDIF}
{$IFDEF Windows}{$APPTYPE CONSOLE}{$ENDIF}
{$DEFINE TIORUN}
uses
sysutils;
const
charSet : array[0..36] of char ='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ';
type
tNumtoBase = record
ntb_dgt : array[0..31-8] of byte;
ntb_cnt,
ntb_bas : Int32;
end;
var
dgtSqrRoot : array[0..31-8] of byte;
sl : array of string;
s2Delta : array of Uint32;
Num,
sqr2B,
deltaSqrNum : tNumtoBase;
procedure Conv2num(var num:tNumtoBase;n:Uint64;base:NativeUint);
var
quot :UInt64;
i :NativeUint;
Begin
i := 0;
repeat
quot := n div base;
Num.ntb_dgt[i] := n-quot*base;
n := quot;
inc(i);
until n = 0;
Num.ntb_cnt := i;
Num.ntb_bas := base;
//clear upper digits
For i := i to high(tNumtoBase.ntb_dgt) do
Num.ntb_dgt[i] := 0;
end;
function OutNum(const num:tNumtoBase):AnsiString;
var
i,j : NativeInt;
Begin
with num do
Begin
setlength(result,ntb_cnt);
j := 1;
For i := ntb_cnt-1 downto 0 do
Begin
result[j] := charSet[ntb_dgt[i]];
inc(j);
end;
end;
end;
procedure InsertSolution(i,penHoloCnt : NativeInt);
begin
sl[penHoloCnt] := Format('%s^2 = %s',[OutNum(Num),OutNum(sqr2B)]);
s2delta[penHoloCnt] := i;
{$IFNDEF TIORUN}
//a little action in the terminal
write(penHoloCnt:8,i :10,' ',sl[penHoloCnt],#13);
IF penHoloCnt= 0 then
writeln;
{$ENDIF}
end;
procedure IncNumBig(var add1:tNumtoBase;n:NativeUInt);
//prerequisites
//bases are the same,delta : NativeUint
var
i,s,b,carry : NativeInt;
Begin
b := add1.ntb_bas;
i := 0;
carry := 0;
while n > 0 do
Begin
s := add1.ntb_dgt[i]+carry+ n MOD b;
carry := Ord(s>=b);
s := s- (-carry AND b);
add1.ntb_dgt[i] := s;
n := n div b;
inc(i);
end;
while carry <> 0 do
Begin
s := add1.ntb_dgt[i]+carry;
carry := Ord(s>=b);
s := s- (-carry AND b);
add1.ntb_dgt[i] := s;
inc(i);
end;
IF add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
procedure IncNum(var add1:tNumtoBase;carry:NativeInt);
//prerequisites: bases are the same, carry==delta < base
var
i,s,b : NativeInt;
Begin
b := add1.ntb_bas;
i := 0;
while carry <> 0 do
Begin
s := add1.ntb_dgt[i]+carry;
carry := Ord(s>=b);
s := s- (-carry AND b);
add1.ntb_dgt[i] := s;
inc(i);
end;
IF add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
procedure AddNum(var add1,add2:tNumtoBase);
//prerequisites
//bases are the same,add1>add2, add1 <= add1+add2;
var
i,carry,s,b : NativeInt;
Begin
b := add1.ntb_bas;
carry := 0;
For i := 0 to add2.ntb_cnt-1 do
begin
s := add1.ntb_dgt[i]+add2.ntb_dgt[i]+carry;
carry := Ord(s>=b);
s := s- (-carry AND b);
add1.ntb_dgt[i] := s;
end;
i := add2.ntb_cnt;
while carry = 1 do
Begin
s := add1.ntb_dgt[i]+carry;
carry := Ord(s>=b);
// remove of if s>b then by bit-twiddling
s := s- (-carry AND b);
add1.ntb_dgt[i] := s;
inc(i);
end;
IF add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
procedure Test(base:NativeUInt);
var
n,penHoloCnt : Uint64;
b1,i,j,TestSet,CheckSet,dgtRoot,step,dNum : NativeInt;
Begin
penHoloCnt := 0;
b1 := Base-1;
dgtRoot := ((Base*b1) DIV 2) MOD b1;
For j := 1 to b1 do
dgtSqrRoot[j] := (j*j) MOD b1;
// square number containing 1,2..,base-1
// now estimate root for 1234..Base-1
// if base is even then root = 111...1 in Base
// if base is odd then root = sqrt(base)*(111...1) in Base
n := 0;
For j := Base DIV 2 downto 1 do
n := n*base+1;
if ODD(BASE) then
n := trunc(n*Sqrt(base));
// increment n til it fits (n*n) MOD b1 = dgtroot
j := B1;
repeat
if sqr(n MOD b1) MOD b1 = dgtroot then
BREAK;
inc(n);
dec(j);
until j = 0;
//calc nxn
Conv2num(Num,n,base);
deltaSqrNum := Num;
Conv2num(sqr2B,0,base);
j := 1;
repeat
if j AND n <> 0 then
AddNum(sqr2B,deltaSqrNum);
AddNum(deltaSqrNum,deltaSqrNum);
j +=j;
until j > n;
// calc step size i
i := 0;
For j := 1 to Base-1 do
if dgtSqrRoot[j] = dgtRoot then
inc(i);
// if i stays 0 than an extra digit will be needed -> no penholodigital
if i <> 0 then
Begin
step := b1 DIV i;
Writeln('Step size ',step,' decimal : ',n,' in base: ',OutNum(Num),' ',OutNum(sqr2B));
// calc delta of square numbers for incremnt n by step
Conv2num(deltaSqrNum,2*step*n,base);
IncNumBig(deltaSqrNum,step*step);
dNum := 2*step*step;
//all digits without 0
CheckSet := 0;
For j := b1 downto 1 do
CheckSet := CheckSet OR (1 shl j);
i := 0;
repeat
//count used digits
TestSet := 0;
For j := sqr2B.ntb_cnt-1 downto 0 do
TestSet := TestSet OR (1 shl sqr2B.ntb_dgt[j]);
IF CheckSet=TestSet then
Begin
//now correct number
IncNumBig(num,i*step);
InsertSolution(i,penHoloCnt);
inc(penHoloCnt);
i := 0;
end;
//next square number
AddNum(sqr2B,deltaSqrNum);
IncNumBig(deltaSqrNum,dNum);// dnum mostly base
inc(i);
until sqr2B.ntb_cnt >= base;
end;
if i = 0 then
Begin
Writeln('Penholodigital squares in base: ',base:2,' are not possible');
writeln;
EXIT;
end
else
Writeln('There are a total of ',penHoloCnt,' penholodigital squares in base: ',base:2);
if (penHoloCnt > 0) AND (base < 11) then
begin
j := 0;
while penHoloCnt-j > 3 do
begin
writeln(sl[j],',',sl[j+1],',',sl[j+2]);
inc(j,3);
end;
write(sl[j]);
For j := j+1 to penHoloCnt-1 do
write(',',sl[j]);
writeln;
end
else
if penHoloCnt > 1 then
begin
writeln(sl[0],',',sl[penHoloCnt-1]);
end;
writeln;
end;
var
T0: TDateTime;
base :nativeInt;
begin
T0 := now;
setlength(sl,51160);
setlength(s2Delta,51160);
For base := 19 to 19 do
Test(base);
writeln('Total runtime in s ',(now-T0)*86400:10:3);
{$IFDEF WINDOWS}readln;{$ENDIF}
end.
- @TIO.RUN:
Step size 1 decimal : 1 in base: 1 1 There are a total of 1 penholodigital squares in base: 2 1^2 = 1 Step size 2 decimal : 1 in base: 1 1 There are a total of 0 penholodigital squares in base: 3 Step size 3 decimal : 6 in base: 12 210 There are a total of 0 penholodigital squares in base: 4 Penholodigital squares in base: 5 are not possible Step size 5 decimal : 45 in base: 113 13213 There are a total of 2 penholodigital squares in base: 6 122^2 = 15324,221^2 = 53241 Step size 6 decimal : 153 in base: 306 125151 There are a total of 1 penholodigital squares in base: 7 645^2 = 623514 Step size 7 decimal : 588 in base: 1114 1243220 There are a total of 1 penholodigital squares in base: 8 2453^2 = 6532471 Step size 4 decimal : 2462 in base: 3335 12357657 There are a total of 10 penholodigital squares in base: 9 3825^2 = 16328547,3847^2 = 16523874,4617^2 = 23875614 4761^2 = 25487631,6561^2 = 47865231,6574^2 = 48162537 6844^2 = 53184267,7285^2 = 58624317,7821^2 = 68573241 8554^2 = 82314657 Step size 3 decimal : 11112 in base: 11112 123476544 There are a total of 30 penholodigital squares in base: 10 11826^2 = 139854276,12363^2 = 152843769,12543^2 = 157326849 14676^2 = 215384976,15681^2 = 245893761,15963^2 = 254817369 18072^2 = 326597184,19023^2 = 361874529,19377^2 = 375468129 19569^2 = 382945761,19629^2 = 385297641,20316^2 = 412739856 22887^2 = 523814769,23019^2 = 529874361,23178^2 = 537219684 23439^2 = 549386721,24237^2 = 587432169,24276^2 = 589324176 24441^2 = 597362481,24807^2 = 615387249,25059^2 = 627953481 25572^2 = 653927184,25941^2 = 672935481,26409^2 = 697435281 26733^2 = 714653289,27129^2 = 735982641,27273^2 = 743816529 29034^2 = 842973156,29106^2 = 847159236,30384^2 = 923187456 Step size 10 decimal : 53415 in base: 3714A 1234599011 There are a total of 20 penholodigital squares in base: 11 42045^2 = 165742A893,A3A39^2 = 983251A764 Step size 11 decimal : 271458 in base: 111116 12346543230 There are a total of 23 penholodigital squares in base: 12 117789^2 = 135B7482A69,319A37^2 = 9B2573468A1 Penholodigital squares in base: 13 are not possible Step size 13 decimal : 8108737 in base: 1111117 1234576543237 There are a total of 160 penholodigital squares in base: 14 1129535^2 = 126A84D79C53B,3A03226^2 = DB3962A7541C8 Step size 14 decimal : 47266835 in base: 4239EC5 123456E806A71A There are a total of 419 penholodigital squares in base: 15 4240C58^2 = 12378DA5B6EC94,EE25E4A^2 = ED4C93285671BA Step size 15 decimal : 286331160 in base: 11111118 123456876543240 There are a total of 740 penholodigital squares in base: 16 11156EB6^2 = 123DA7F85BCE964,3FD8F786^2 = FEC81B69573DA24 Total runtime in s 7.159 @home 1.573 s
// Running Base 18 and 19 @home Step size 17 decimal : 11668193559 in base: 111111119 12345679876543249 There are a total of 5116 penholodigital squares in base: 18 11150FC0G^2 = 123CD8ABH5G79F6E4,44422DD18^2 = HGEF25738D496BC1A Step size 6 decimal : 78142392501 in base: 4B7ICE111 12345678BB48EGB321 There are a total of 47677 penholodigital squares in base: 19 4B802235A^2 = 1234978CIBD6GFHEA5,II844BIA2^2 = IHG8F71C6DA59BE324 Total runtime in s 1151.714 real 19m11.715s user 19m10.257s sys 0m0.042s
Perl
use v5.36;
use List::Util 'max';
use ntheory <fromdigits todigitstring>;
sub table ($c, @V) { my $t = $c * (my $w = 2 + max map { length } @V); ( sprintf( ('%'.$w.'s')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }
for my $base (9 .. 12) {
my $testr = reverse my $test = substr('123456789abcdef',0,$base-1);
my $start = int sqrt fromdigits($test, $base);
my $end = int sqrt fromdigits($testr, $base);
my @nums = grep { $test eq join '', sort split '', todigitstring($_**2, $base) } $start .. $end;
printf "There are a total of %d penholodigital squares in base $base:\n", scalar @nums;
say table 4, map { todigitstring($_,$base) . '² = ' . todigitstring($_**2,$base) } @nums;
}
- Output:
There are a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There are a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There are a total of 20 penholodigital squares in base 11: 42045² = 165742a893 43152² = 173a652894 44926² = 18792a6453 47149² = 1a67395824 47257² = 1a76392485 52071² = 249a758631 54457² = 2719634a85 55979² = 286a795314 59597² = 314672a895 632a4² = 3671a89245 64069² = 376198a254 68335² = 41697528a3 71485² = 46928a7153 81196² = 5a79286413 83608² = 632a741859 86074² = 6713498a25 89468² = 7148563a29 91429² = 76315982a4 93319² = 795186a234 a3a39² = 983251a764 There are a total of 23 penholodigital squares in base 12: 117789² = 135b7482a69 16357b² = 23a5b976481 16762b² = 24ab5379861 16906b² = 25386749ba1 173434² = 26b859a3714 178278² = 2835ba17694 1a1993² = 34a8125b769 1a3595² = 354a279b681 1b0451² = 3824b7569a1 1b7545² = 3a5b2487961 2084a9² = 42a1583b769 235273² = 5287ba13469 2528b5² = 5b23a879641 25b564² = 62937b5a814 262174² = 63a8527b194 285a44² = 73b615a8294 29a977² = 7b9284a5361 2a7617² = 83ab5479261 2b0144² = 8617b35a294 307381² = 93825a67b41 310828² = 96528ab7314 319488² = 9ab65823714 319a37² = 9b2573468a1
Phix
using string maths (and not particularly fast)
with javascript_semantics
procedure penholodigital(integer base)
sequence penholodigitals = {}
-- search squares of n from eg for base=4 sqrt(123)
-- (and stop when next square has too many digits)
atom n = 0, t1 = time()+1, carry = 0
for d=1 to base-1 do n = n*base+d end for
n = ceil(sqrt(n)) assert(integer(n))
string digits = "123456789ABCDEF"[1..base-1],
square = sprintf("%A",{{base,n*n}})
integer div = prime_factors(base-1,true)[$]
bool ndz = mod(n,div)=0
while true do -- (until we get a carry off the end of square)
if ndz and not find('0',square) and sort(square)=digits then
penholodigitals &= {{{base,n},square}} -- (fmt later)
end if
-- add 2n+1 to get the next square (using string maths)
carry += 2*n+1
n += 1
ndz = mod(n,div)=0
if ndz then
integer d = base-1
while carry and d do
atom ch = square[d]
ch += carry - iff(ch<='9'?'0':'A'-10)
carry = floor(ch/base)
ch = remainder(ch,base)
square[d] = ch + iff(ch<=9?'0':'A'-10)
d -= 1
end while
if carry then exit end if
if time()>t1 then
integer l = length(penholodigitals),
e = {1,0,0,0,2,1,1,10,30,20,23,0,160,419,740}[base-1]
progress("scanning base %d, %s (%d/%d found)\r",{base,square,l,e})
t1 = time()+1
end if
end if
end while
progress("")
integer count = length(penholodigitals)
if base>12 and count>2 then
penholodigitals = extract(penholodigitals,{1,-1})
end if
penholodigitals = apply(true,sprintf,{{"%A**2 = %s"},penholodigitals})
string p = iff(count?":\n"&iff(base>12?sprintf("%s .. %s\n",penholodigitals)
:join_by(penholodigitals,1,3)):"\n"),
{is,s} = iff(count=1?{"is",""}:{"are","s"})
printf(1,"There %s %d penholodigital%s in base %d%s\n",{is,count,s,base,p})
end procedure
integer lim = iff(platform()=JS?14: -- ~18s
iff(machine_bits()=32?15 -- (initial square(16) is 4 out)
:16)) -- ~7.5 mins (ho hum)
papply(tagset(lim,2),penholodigital)
- Output:
There is 1 penholodigital in base 2: 1**2 = 1 There are 0 penholodigitals in base 3 There are 0 penholodigitals in base 4 There are 0 penholodigitals in base 5 There are 2 penholodigitals in base 6: 122**2 = 15324 221**2 = 53241 There is 1 penholodigital in base 7: 645**2 = 623514 There is 1 penholodigital in base 8: 2453**2 = 6532471 There are 10 penholodigitals in base 9: 3825**2 = 16328547 3847**2 = 16523874 4617**2 = 23875614 4761**2 = 25487631 6561**2 = 47865231 6574**2 = 48162537 6844**2 = 53184267 7285**2 = 58624317 7821**2 = 68573241 8554**2 = 82314657 There are 30 penholodigitals in base 10: 11826**2 = 139854276 12363**2 = 152843769 12543**2 = 157326849 14676**2 = 215384976 15681**2 = 245893761 15963**2 = 254817369 18072**2 = 326597184 19023**2 = 361874529 19377**2 = 375468129 19569**2 = 382945761 19629**2 = 385297641 20316**2 = 412739856 22887**2 = 523814769 23019**2 = 529874361 23178**2 = 537219684 23439**2 = 549386721 24237**2 = 587432169 24276**2 = 589324176 24441**2 = 597362481 24807**2 = 615387249 25059**2 = 627953481 25572**2 = 653927184 25941**2 = 672935481 26409**2 = 697435281 26733**2 = 714653289 27129**2 = 735982641 27273**2 = 743816529 29034**2 = 842973156 29106**2 = 847159236 30384**2 = 923187456 There are 20 penholodigitals in base 11: 42045**2 = 165742A893 43152**2 = 173A652894 44926**2 = 18792A6453 47149**2 = 1A67395824 47257**2 = 1A76392485 52071**2 = 249A758631 54457**2 = 2719634A85 55979**2 = 286A795314 59597**2 = 314672A895 632A4**2 = 3671A89245 64069**2 = 376198A254 68335**2 = 41697528A3 71485**2 = 46928A7153 81196**2 = 5A79286413 83608**2 = 632A741859 86074**2 = 6713498A25 89468**2 = 7148563A29 91429**2 = 76315982A4 93319**2 = 795186A234 A3A39**2 = 983251A764 There are 23 penholodigitals in base 12: 117789**2 = 135B7482A69 16357B**2 = 23A5B976481 16762B**2 = 24AB5379861 16906B**2 = 25386749BA1 173434**2 = 26B859A3714 178278**2 = 2835BA17694 1A1993**2 = 34A8125B769 1A3595**2 = 354A279B681 1B0451**2 = 3824B7569A1 1B7545**2 = 3A5B2487961 2084A9**2 = 42A1583B769 235273**2 = 5287BA13469 2528B5**2 = 5B23A879641 25B564**2 = 62937B5A814 262174**2 = 63A8527B194 285A44**2 = 73B615A8294 29A977**2 = 7B9284A5361 2A7617**2 = 83AB5479261 2B0144**2 = 8617B35A294 307381**2 = 93825A67B41 310828**2 = 96528AB7314 319488**2 = 9AB65823714 319A37**2 = 9B2573468A1 There are 0 penholodigitals in base 13 There are 160 penholodigitals in base 14: 1129535**2 = 126A84D79C53B .. 3A03226**2 = DB3962A7541C8 There are 419 penholodigitals in base 15: 4240C58**2 = 12378DA5B6EC94 .. EE25E4A**2 = ED4C93285671BA There are 740 penholodigitals in base 16: 11156EB6**2 = 123DA7F85BCE964 .. 3FD8F786**2 = FEC81B69573DA24
slightly faster
Using string digit-wise maths (and still not particularly fast). Same output as above, but 32 bit can also do base 16.
with javascript_semantics
--
-- We can check for the presence of any zeroes or duplicate digits as we go.
-- (Please gloss over the gaff of increasing #digits, which makes the next a rather daft example.)
-- We can also leave partial results unfinished, for (a bad) example when adding 199 = 2*99+1 to go
-- from 99*99 = {9,8,0,1} to 100*100 = {1,0,0,0,0} we can simply stop when we see a 0/dup in the
-- finished part of the result, on {9,8,20,0}, and it won't hurt us at all when adding 201 = 2*100+1
-- to get the next square. That does mean we need to be a little smarter displaying progress, though.
-- We can also utilise a partial to avoid overflow and get 16 digits to work properly on 32-bit.
-- We can also skip +199 if that ain't gona work and +400 on the next iteration instead, iyswim.
-- We can also use a trivial integer-sized bitmask to check for duplicate digits, which won't hit any
-- problems (on Phix) up to at least 29 digits, or 61 digits on 64-bit, and start off with bitmask=1
-- to effectively mean "0 already seen", so that test no longer needs to be a separate check.
-- Combined, these optimisations improved things from 7 minutes 23s to 4 mins 15s, or about 43%.
-- Partials only saved 45s, or ~10%, admittedly rather less than hoped for, but still something.
--
function aleph(sequence s, integer base)
-- convert eg {1,15} to "1F"
-- s may be partial: show digits>=base as '?'
string res = repeat(' ',length(s))
for i,d in s do
res[i] = iff(d>=base?'?':d+iff(d<=9?'0':'A'-10))
end for
return res
end function
function reformat(sequence rs)
sequence {r,s} = rs
string root = sprintf("%A",{r}),
square = aleph(s,r[1])
return sprintf("%s**2 = %s",{root,square})
end function
procedure penholodigital(integer base)
-- search squares of n from eg sqrt(123) for base=4
-- (and stop when next square has too many digits)
sequence penholodigitals = {}
atom n = 0, t1 = time()+1, carry = 0
for d=1 to base-1 do n = n*base+d end for -- (eg 123)
n = ceil(sqrt(n))-1 assert(integer(n))
-- (^^ -1 as we bump square and set zod on first iter)
sequence square = repeat(0,base-1)
integer div = prime_factors(base-1,true)[$], d, dn = n, mbit
-- square[$] = n*n -- 9 out for base of 16 on 32 bit, so:
-- for eg 123*123 start off with {1*123,2*123,3*123}
for d = base-1 to 1 by -1 do
square[d] = remainder(dn,base)*n
dn = floor(dn/base)
end for
bool ndz = mod(n,div)=0,
zod -- zero or duplicate digit flag
while true do -- (until we get a carry off the end of square)
-- add 2n+1 to get the next square (using digitwise maths)
carry += 2*n+1
n += 1
ndz = mod(n,div)=0
if ndz then
zod = false
integer mask = 01 -- (treat 0s as dups)
d = base-1
while carry and d do
atom ch = square[d] + carry
carry = floor(ch/base)
ch = remainder(ch,base)
square[d] = ch
d -= 1
mbit = power(2,ch)
if and_bits(mask,mbit) then
zod = true -- duplicate digit (or 0)
if d then
square[d] += carry
carry = 0
-- theoretically better, but in practice worse:
-- carry = carry>((base-square[1])*power(base,d-1))
end if
exit
end if
mask += mbit
end while
if carry then exit end if
if time()>t1 then
integer l = length(penholodigitals),
-- mini-cheat for the progress messages only:
e = {1,0,0,0,2,1,1,10,30,20,23,0,160,419,740}[base-1]
string sq = aleph(square,base)
progress("scanning base %d, %s (%d/%d found)\r",{base,sq,l,e})
t1 = time()+1
end if
if not zod then
while d do -- finish zod checks & clean up any "legacy" partials
atom ch = square[d]
d -= 1
if ch>=base then
carry = floor(ch/base)
ch = remainder(ch,base) -- (nb: needed for mbit below)
square[d+1] = ch
if d=0 then exit end if -- (and any carry quits)
square[d] += carry
carry = 0
end if
mbit = power(2,ch)
if and_bits(mask,mbit) then
zod = true -- duplicate digit (or 0)
exit
end if
mask += mbit
end while
if carry then exit end if
if not zod then
penholodigitals &= {{{base,n},deep_copy(square)}} -- (fmt later)
end if
end if
end if
end while
progress("")
integer count = length(penholodigitals)
if base>12 and count>2 then
penholodigitals = extract(penholodigitals,{1,-1})
end if
penholodigitals = apply(penholodigitals,reformat)
string p = iff(count?":\n"&iff(base>12?sprintf("%s .. %s\n",penholodigitals)
:join_by(penholodigitals,1,3)):"\n"),
{is,s} = iff(count=1?{"is",""}:{"are","s"})
printf(1,"There %s %d penholodigital%s in base %d%s\n",{is,count,s,base,p})
end procedure
integer lim = iff(platform()=JS?14:16) -- js ~3.8s, desktop(64/32bit) 4/6 mins
papply(tagset(lim,2),penholodigital)
Javascript is actually nearly twice as fast as desktop/32bit, finishing base 16 in 3 mins 35s (but blank screen till then)
Raku
(9 .. 12).map: -> $base {
my $test = (1 ..^ $base)».base($base).join;
my $start = $test .parse-base($base).sqrt.Int;
my $end = $test.flip.parse-base($base).sqrt.Int;
say "\nThere is a total of {+$_} penholodigital squares in base $base:\n" ~
.map({"{.base($base)}² = {.².base($base)}"}).batch(3)».join(", ").join: "\n" given
($start .. $end).grep: *².base($base).comb.sort.join eq $test
}
(13 .. 16).hyper(:1batch).map: -> $base {
my $test = (1 ..^ $base)».base($base).join;
my $start = $test .parse-base($base).sqrt.Int;
my $end = $test.flip.parse-base($base).sqrt.Int;
my @penholo = ($start .. $end).grep: *².base($base).comb.sort.join eq $test;
say "\nThere is a total of {+@penholo} penholodigital squares in base $base:";
say @penholo[0,*-1].map({"{.base($base)}² = {.².base($base)}"}).batch(3)».join(", ").join: "\n" if +@penholo;
}
- Output:
There is a total of 10 penholodigital squares in base 9: 3825² = 16328547, 3847² = 16523874, 4617² = 23875614 4761² = 25487631, 6561² = 47865231, 6574² = 48162537 6844² = 53184267, 7285² = 58624317, 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276, 12363² = 152843769, 12543² = 157326849 14676² = 215384976, 15681² = 245893761, 15963² = 254817369 18072² = 326597184, 19023² = 361874529, 19377² = 375468129 19569² = 382945761, 19629² = 385297641, 20316² = 412739856 22887² = 523814769, 23019² = 529874361, 23178² = 537219684 23439² = 549386721, 24237² = 587432169, 24276² = 589324176 24441² = 597362481, 24807² = 615387249, 25059² = 627953481 25572² = 653927184, 25941² = 672935481, 26409² = 697435281 26733² = 714653289, 27129² = 735982641, 27273² = 743816529 29034² = 842973156, 29106² = 847159236, 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742A893, 43152² = 173A652894, 44926² = 18792A6453 47149² = 1A67395824, 47257² = 1A76392485, 52071² = 249A758631 54457² = 2719634A85, 55979² = 286A795314, 59597² = 314672A895 632A4² = 3671A89245, 64069² = 376198A254, 68335² = 41697528A3 71485² = 46928A7153, 81196² = 5A79286413, 83608² = 632A741859 86074² = 6713498A25, 89468² = 7148563A29, 91429² = 76315982A4 93319² = 795186A234, A3A39² = 983251A764 There is a total of 23 penholodigital squares in base 12: 117789² = 135B7482A69, 16357B² = 23A5B976481, 16762B² = 24AB5379861 16906B² = 25386749BA1, 173434² = 26B859A3714, 178278² = 2835BA17694 1A1993² = 34A8125B769, 1A3595² = 354A279B681, 1B0451² = 3824B7569A1 1B7545² = 3A5B2487961, 2084A9² = 42A1583B769, 235273² = 5287BA13469 2528B5² = 5B23A879641, 25B564² = 62937B5A814, 262174² = 63A8527B194 285A44² = 73B615A8294, 29A977² = 7B9284A5361, 2A7617² = 83AB5479261 2B0144² = 8617B35A294, 307381² = 93825A67B41, 310828² = 96528AB7314 319488² = 9AB65823714, 319A37² = 9B2573468A1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126A84D79C53B, 3A03226² = DB3962A7541C8 There is a total of 419 penholodigital squares in base 15: 4240C58² = 12378DA5B6EC94, EE25E4A² = ED4C93285671BA There is a total of 740 penholodigital squares in base 16: 11156EB6² = 123DA7F85BCE964, 3FD8F786² = FEC81B69573DA24
RPL
Code | Comments |
---|---|
"0123456789ABCDEF" 'Digits' STO ≪ → base ≪ "" SWAP WHILE DUP REPEAT DUP base MOD Digits OVER 1 + DUP SUB 4 ROLL + ROT ROT - base / RND END DROP ≫ ≫ 'D→BAS' STO ≪ → base ≪ 0 1 SIZE FOR j base * OVER j j SUB Digits SWAP POS 1 - + NEXT SWAP DROP ≫ ≫ 'BAS→D' STO ≪ → number base digits ≪ IF number "0" POS THEN 0 ELSE {} base + 0 CON 1 1 PUT 1 number SIZE FOR j Digits number j DUP SUB POS 1 PUT NEXT CNRM base == END ≫ ≫ 'PHD?' STO ≪ 0 0 → base first last ≪ 0 1 SF Digits 2 base SUB base BAS→D√ FLOOR "FEDCBA987654321" 17 base - 15 SUB base BAS→D√ CEIL FOR n n SQ base D→BAS IF base PHD? THEN 1 + n 'last' STO IF 1 FS?C THEN n 'first' STO END END NEXT IF first THEN first base D→BAS "^2 = " + first SQ base D→BAS + last base D→BAS "^2 = " + last SQ base D→BAS + END ≫ ≫ 'PHDSQ' STO |
Constant ( n base -- "###" ) ( "###" base -- n ) ( "###" base -- boolean ) Initialize variables o/w an array of counters the sum of counters must equal base ( base -- #PHD "first PHD" "last PHD" ) start search with "123..b" end search with "b..321" Display detailed results |
The following lines of command deliver what is required:
9 PHDSQ 10 PHDSQ 11 PHDSQ 12 PHDSQ
- Output:
12: 10 11: "3825^2 = 16328547" 10: "8554^2 = 82314657" 9: 30 8: "11826^2 = 139854276" 7: "30384^2 = 923187456" 6: 20 5: "42045^2 = 165742A893" 4: "A3A39^2 = 983251A764" 3: 23 2: "117789^2 = 135B7482A69" 1: "319A37^2 = 9B2573468A1"
Wren
This is limited to base 14 as base 15 would overflow Wren's safe integer limit of 2^53.
Although I'm not quite sure why, it appears that a necessary condition for a number to be a penholodigital square is for its square root to be exactly divisible by the highest prime factor of (base - 1).
import "./math" for Int
import "./fmt" for Conv, Fmt
var primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
var digits = "123456789ABCD"
for (b in 9..14) {
var master = 1
for (d in 1...b) master = master * primes[d-1]
var phd = []
var min = Conv.atoi(digits[0..(b-2)], b).sqrt.ceil
var max = Conv.atoi(digits[(b-2)..0], b).sqrt.floor
var div = Int.primeFactors(b-1)[-1]
for (i in min..max) {
if ((i % div) != 0) continue
var sq = i * i
var digs = Int.digits(sq, b)
if (digs.contains(0)) continue
var key = 1
for (dig in digs) key = key * primes[dig-1]
if (key == master) phd.add(i)
}
System.print("There is a total of %(phd.count) penholodigital squares in base %(b):")
if (b > 13) phd = [phd[0], phd[-1]]
for (i in 0...phd.count) {
Fmt.write("$s² = $s ", Conv.Itoa(phd[i], b), Conv.Itoa(phd[i] * phd[i], b))
if ((i + 1) % 3 == 0) System.print()
}
if (phd.count % 3 != 0) System.print()
System.print()
}
- Output:
There is a total of 10 penholodigital squares in base 9: 3825² = 16328547 3847² = 16523874 4617² = 23875614 4761² = 25487631 6561² = 47865231 6574² = 48162537 6844² = 53184267 7285² = 58624317 7821² = 68573241 8554² = 82314657 There is a total of 30 penholodigital squares in base 10: 11826² = 139854276 12363² = 152843769 12543² = 157326849 14676² = 215384976 15681² = 245893761 15963² = 254817369 18072² = 326597184 19023² = 361874529 19377² = 375468129 19569² = 382945761 19629² = 385297641 20316² = 412739856 22887² = 523814769 23019² = 529874361 23178² = 537219684 23439² = 549386721 24237² = 587432169 24276² = 589324176 24441² = 597362481 24807² = 615387249 25059² = 627953481 25572² = 653927184 25941² = 672935481 26409² = 697435281 26733² = 714653289 27129² = 735982641 27273² = 743816529 29034² = 842973156 29106² = 847159236 30384² = 923187456 There is a total of 20 penholodigital squares in base 11: 42045² = 165742A893 43152² = 173A652894 44926² = 18792A6453 47149² = 1A67395824 47257² = 1A76392485 52071² = 249A758631 54457² = 2719634A85 55979² = 286A795314 59597² = 314672A895 632A4² = 3671A89245 64069² = 376198A254 68335² = 41697528A3 71485² = 46928A7153 81196² = 5A79286413 83608² = 632A741859 86074² = 6713498A25 89468² = 7148563A29 91429² = 76315982A4 93319² = 795186A234 A3A39² = 983251A764 There is a total of 23 penholodigital squares in base 12: 117789² = 135B7482A69 16357B² = 23A5B976481 16762B² = 24AB5379861 16906B² = 25386749BA1 173434² = 26B859A3714 178278² = 2835BA17694 1A1993² = 34A8125B769 1A3595² = 354A279B681 1B0451² = 3824B7569A1 1B7545² = 3A5B2487961 2084A9² = 42A1583B769 235273² = 5287BA13469 2528B5² = 5B23A879641 25B564² = 62937B5A814 262174² = 63A8527B194 285A44² = 73B615A8294 29A977² = 7B9284A5361 2A7617² = 83AB5479261 2B0144² = 8617B35A294 307381² = 93825A67B41 310828² = 96528AB7314 319488² = 9AB65823714 319A37² = 9B2573468A1 There is a total of 0 penholodigital squares in base 13: There is a total of 160 penholodigital squares in base 14: 1129535² = 126A84D79C53B 3A03226² = DB3962A7541C8
XPL0
Real, 64-bit numbers provide the precision needed beyond 32-bit integers to reach base 14 (14^13 = 7.94e14, which is 49.5 bits). Runs in 37 seconds on Pi4.
real Base; \Number Base being used [9..14]
proc NumOut(N); \Display N in the specified Base
real N;
int Remain;
[Remain:= fix(Mod(N, Base));
N:= Floor(N/Base);
if N # 0. then NumOut(N);
ChOut(0, Remain + (if Remain <= 9 then ^0 else ^A-10));
];
proc ShowPenSq(N); \Display N = N^2
real N;
[NumOut(N);
Text(0, "^^2 = ");
NumOut(N*N);
Text(0, " ");
];
func Penholodigital(N); \Return 'true' if N is penholodigital
real N;
int Used, Remain;
[Used:= 1; \can't contain 0
while N # 0. do
[Remain:= fix(Mod(N, Base));
N:= Floor(N/Base);
if Used & 1<<Remain then return false;
Used:= Used ! 1<<Remain;
];
return Used = 1<<fix(Base) - 1;
];
int Cnt;
real N, Limit, FirstN, LastN;
[Base:= 9.;
repeat Cnt:= 0; FirstN:= 0.;
N:= Floor(Sqrt(Pow(Base, Base-2.)));
Limit:= sqrt(Pow(Base, Base-1.));
repeat if Penholodigital(N*N) then
[Cnt:= Cnt+1;
if FirstN = 0. then FirstN:= N;
if Base <= 12. then
[ShowPenSq(N);
if rem(Cnt/3) = 0 then CrLf(0);
]
else LastN:= N;
];
N:= N + 1.;
until N >= Limit;
if rem(Cnt/3) # 0 then CrLf(0);
Text(0, "There are "); IntOut(0, Cnt);
Text(0, " penholodigital squares in base ");
IntOut(0, fix(Base)); CrLf(0);
if Base >= 13. and Cnt > 0 then
[ShowPenSq(FirstN);
ShowPenSq(LastN);
CrLf(0);
];
CrLf(0);
Base:= Base + 1.;
until Base >= 15.;
]
- Output:
3825^2 = 16328547 3847^2 = 16523874 4617^2 = 23875614 4761^2 = 25487631 6561^2 = 47865231 6574^2 = 48162537 6844^2 = 53184267 7285^2 = 58624317 7821^2 = 68573241 8554^2 = 82314657 There are 10 penholodigital squares in base 9 11826^2 = 139854276 12363^2 = 152843769 12543^2 = 157326849 14676^2 = 215384976 15681^2 = 245893761 15963^2 = 254817369 18072^2 = 326597184 19023^2 = 361874529 19377^2 = 375468129 19569^2 = 382945761 19629^2 = 385297641 20316^2 = 412739856 22887^2 = 523814769 23019^2 = 529874361 23178^2 = 537219684 23439^2 = 549386721 24237^2 = 587432169 24276^2 = 589324176 24441^2 = 597362481 24807^2 = 615387249 25059^2 = 627953481 25572^2 = 653927184 25941^2 = 672935481 26409^2 = 697435281 26733^2 = 714653289 27129^2 = 735982641 27273^2 = 743816529 29034^2 = 842973156 29106^2 = 847159236 30384^2 = 923187456 There are 30 penholodigital squares in base 10 42045^2 = 165742A893 43152^2 = 173A652894 44926^2 = 18792A6453 47149^2 = 1A67395824 47257^2 = 1A76392485 52071^2 = 249A758631 54457^2 = 2719634A85 55979^2 = 286A795314 59597^2 = 314672A895 632A4^2 = 3671A89245 64069^2 = 376198A254 68335^2 = 41697528A3 71485^2 = 46928A7153 81196^2 = 5A79286413 83608^2 = 632A741859 86074^2 = 6713498A25 89468^2 = 7148563A29 91429^2 = 76315982A4 93319^2 = 795186A234 A3A39^2 = 983251A764 There are 20 penholodigital squares in base 11 117789^2 = 135B7482A69 16357B^2 = 23A5B976481 16762B^2 = 24AB5379861 16906B^2 = 25386749BA1 173434^2 = 26B859A3714 178278^2 = 2835BA17694 1A1993^2 = 34A8125B769 1A3595^2 = 354A279B681 1B0451^2 = 3824B7569A1 1B7545^2 = 3A5B2487961 2084A9^2 = 42A1583B769 235273^2 = 5287BA13469 2528B5^2 = 5B23A879641 25B564^2 = 62937B5A814 262174^2 = 63A8527B194 285A44^2 = 73B615A8294 29A977^2 = 7B9284A5361 2A7617^2 = 83AB5479261 2B0144^2 = 8617B35A294 307381^2 = 93825A67B41 310828^2 = 96528AB7314 319488^2 = 9AB65823714 319A37^2 = 9B2573468A1 There are 23 penholodigital squares in base 12 There are 0 penholodigital squares in base 13 There are 160 penholodigital squares in base 14 1129535^2 = 126A84D79C53B 3A03226^2 = DB3962A7541C8