# Penholodigital squares

Penholodigital squares is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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.

From the Latin prefix pene- (before, or next to, nearly)

and holo- (whole, or all)

penholodigital: Nearly-all-digits.

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

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, ... ?

## 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

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)

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"

## Delphi

Works with: Delphi version 6.0

{Library routine included here for clarity}

{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
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}
{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? }
finally SL.Free; end;
end;

{Get smallest N-digit number of specified radix}
begin
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
end;

{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}
{Scan all the number for Penholodigital squares}
I:=Start;
while I<=Stop do
begin
begin
Inc(Cnt);
if I<Stop then S:=S+'   ';
If (Cnt mod 3)=0 then S:=S+CRLF;
end;
Inc(I);
end;
H:=H+'Count='+IntToStr(Cnt)+CRLF;
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
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

Translation of: XPLo
#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

Translation of: Wren
Library: Go-rcu
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
}}

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'
}}

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
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
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
23 penholodigital squares in base 12
135b7482a69=117789² 23a5b976481=16357 24ab5379861=16762 25386749ba1=16906 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...

## jq

Works with: jq

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];

# 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 );

range(a;b) as \$base
| penholodigital(\$base)
| "\n\nThere are \(length) penholodigital squares in base \(\$base):",
(_nwise(3)
| map("\(.[0])² = \(.[1])" )
| join("  "));

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

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)
hi, lo = isqrt(evalpoly(base, 1:base-1)), isqrt(evalpoly(base, base-1:-1:1))
digits!(dig, n * n; base)
0 in dig && continue
if all(i -> count(==(i), dig) == 1, 1:base-1)
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 of: Wren

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:
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):

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;

//prerequisites
//bases are the same,delta : NativeUint
var
i,s,b,carry : NativeInt;
Begin
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);
n := n div b;
inc(i);
end;

while carry <> 0 do
Begin
carry := Ord(s>=b);
s := s- (-carry AND b);
inc(i);
end;

end;

//prerequisites: bases are the same, carry==delta < base
var
i,s,b : NativeInt;
Begin
i := 0;
while carry <> 0 do
Begin
carry := Ord(s>=b);
s := s- (-carry AND b);
inc(i);
end;
end;

//prerequisites
var
i,carry,s,b : NativeInt;
Begin
carry := 0;
For i := 0 to add2.ntb_cnt-1 do
begin
carry := Ord(s>=b);
s := s- (-carry AND b);
end;

while carry = 1 do
Begin
carry := Ord(s>=b);
// remove of if s>b then by bit-twiddling
s := s- (-carry AND b);
inc(i);
end;

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
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
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);
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

Library: ntheory
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)
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
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)
zod = true  -- duplicate digit (or 0)
exit
end if
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

Works with: Halcyon Calc version 4.2.7
"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

Library: Wren-math
Library: Wren-fmt

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]
}
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