Colorful numbers: Difference between revisions
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dig = digits(n, base=base) |
dig = digits(n, base=base) |
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(1 in dig || 0 in dig || !allunique(dig)) && return false |
(1 in dig || 0 in dig || !allunique(dig)) && return false |
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products = |
products = Set(dig) |
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for i in 2:length(dig), j in 1:length(dig)-i+1 |
for i in 2:length(dig), j in 1:length(dig)-i+1 |
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p = prod(dig[j:j+i-1]) |
p = prod(dig[j:j+i-1]) |
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Revision as of 07:31, 24 February 2022
A colorful number is a non-negative base 10 integer where the product of every sub group of consecutive digits is unique.
- E.G.
24753 is a colorful number. 2, 4, 7, 5, 3, (2×4)8, (4×7)28, (7×5)35, (5×3)15, (2×4×7)56, (4×7×5)140, (7×5×3)105, (2×4×7×5)280, (4×7×5×3)420, (2×4×7×5×3)840
Every product is unique.
2346 is not a colorful number. 2, 3, 4, 6, (2×3)6, (3×4)12, (4×6)24, (2×3×4)48, (3×4×6)72, (2×3×4×6)144
The product 6 is repeated.
Single digit numbers are considered to be colorful. A colorful number larger than 9 cannot contain a repeated digit, the digit 0 or the digit 1. As a consequence, there is a firm upper limit for colorful numbers; no colorful number can have more than 8 digits.
- Task
- Write a routine (subroutine, function, procedure, whatever it may be called in your language) to test if a number is a colorful number or not.
- Use that routine to find all of the colorful numbers less than 100.
- Use that routine to find the largest possible colorful number.
- Stretch
- Find and display the count of colorful numbers in each order of magnitude.
- Find and show the total count of all colorful numbers.
Colorful numbers have no real number theory application. They are more a recreational math puzzle than a useful tool.
J
<lang J> colorful=: {{(-:~.);<@(*/\)\. 10 #.inv y}}"0
I.colorful i.100
0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98
C=: I.colorful <.i.1e8 >./C
98746253
(~.,. #/.~) 10 <.@^. C
__ 1
0 9 1 56 2 328 3 1540 4 5514 5 13956 6 21596 7 14256 #C
57256</lang>
(Note that 0, here is a different order of magnitude than 1.)
Julia
<lang julia>function iscolorful(n, base=10)
0 <= n < 10 && return true
dig = digits(n, base=base)
(1 in dig || 0 in dig || !allunique(dig)) && return false
products = Set(dig)
for i in 2:length(dig), j in 1:length(dig)-i+1
p = prod(dig[j:j+i-1])
p in products && return false
push!(products, p)
end
return true
end
function testcolorfuls()
println("Colorful numbers for 1:25, 26:50, 51:75, and 76:100:")
for i in 1:100
iscolorful(i) && print(rpad(i, 5))
i % 25 == 0 && println()
end
csum = 0
for i in 0:7
j, k = i == 0 ? 0 : 10^i, 10^(i+1) - 1
n = count(i -> iscolorful(i), j:k)
csum += n
println("The count of colorful numbers between $j and $k is $n.")
end
println("The total number of colorful numbers is $csum.")
end
testcolorfuls()
</lang>
- Output:
1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98 The count of colorful numbers between 0 and 9 is 10. The count of colorful numbers between 10 and 99 is 56. The count of colorful numbers between 100 and 999 is 328. The count of colorful numbers between 1000 and 9999 is 1540. The count of colorful numbers between 10000 and 99999 is 5514. The count of colorful numbers between 100000 and 999999 is 13956. The count of colorful numbers between 1000000 and 9999999 is 21596. The count of colorful numbers between 10000000 and 99999999 is 14256. The total number of colorful numbers is 57256.
Phix
You can run this online here.
with javascript_semantics function colourful(integer n) if n<10 then return n>=0 end if sequence digits = sq_sub(sprintf("%d",n),'0'), ud = unique(deep_copy(digits)) integer ln = length(digits) if ud[1]<=1 or length(ud)!=ln then return false end if for i=1 to ln-1 do for j=i+1 to ln do atom prod = product(digits[i..j]) if find(prod,ud) then return false end if ud &= prod end for end for return true end function atom t0 = time() sequence cn = apply(true,sprintf,{{"%2d"},filter(tagset(100,0),colourful)}) printf(1,"The %d colourful numbers less than 100 are:\n%s\n", {length(cn),join_by(cn,1,10," ")}) sequence count = repeat(0,8), used = repeat(false,10) integer largestcn = 0 procedure count_colourful(integer taken=0, string n="") if taken=0 then for digit='0' to '9' do integer dx = digit-'0'+1 used[dx] = true count_colourful(iff(digit<'2'?9:1),""&digit) used[dx] = false end for else integer nn = to_integer(n) if colourful(nn) then integer ln = length(n) count[ln] += 1 if nn>largestcn then largestcn = nn end if end if if taken<9 then for digit='2' to '9' do integer dx = digit-'0'+1 if not used[dx] then used[dx] = true count_colourful(taken+1,n&digit) used[dx] = false end if end for end if end if end procedure count_colourful() printf(1,"The largest possible colourful number is: %,d\n\n",largestcn) atom pow = 10 for dc=1 to length(count) do printf(1," %d digit colourful number count: %,6d - %7.3f%%\n", {dc, count[dc], 100*count[dc]/pow}) pow = iff(pow=10?90:pow*10) end for printf(1,"\nTotal colourful numbers: %,d\n", sum(count)) ?elapsed(time()-t0)
- Output:
The 66 colourful numbers less than 100 are: 0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98 The largest possible colourful number is: 98,746,253 1 digit colourful number count: 10 - 100.000% 2 digit colourful number count: 56 - 62.222% 3 digit colourful number count: 328 - 36.444% 4 digit colourful number count: 1,540 - 17.111% 5 digit colourful number count: 5,514 - 6.127% 6 digit colourful number count: 13,956 - 1.551% 7 digit colourful number count: 21,596 - 0.240% 8 digit colourful number count: 14,256 - 0.016% Total colourful numbers: 57,256 "1.9s"
Raku
<lang perl6>sub is-colorful (Int $n) {
return True if 0 <= $n <= 9;
return False if $n.contains(0) || $n.contains(1) || $n < 0;
my @digits = $n.comb;
my %sums = @digits.Bag;
return False if %sums.values.max > 1;
for 2..@digits -> $group {
@digits.rotor($group => 1 - $group).map: { %sums{ [×] $_ }++ }
return False if %sums.values.max > 1;
}
True
}
put "Colorful numbers less than 100:\n" ~ (^100).race.grep( &is-colorful).batch(10)».fmt("%2d").join: "\n";
my ($start, $total) = 23456789, 10;
print "\nLargest magnitude colorful number: "; .put and last if .Int.&is-colorful for $start.flip … $start;
put "\nCount of colorful numbers for each order of magnitude:\n" ~
"1 digit colorful number count: $total - 100%";
for 2..8 {
put "$_ digit colorful number count: ",
my $c = +(flat $start.comb.combinations($_).map: {.permutations».join».Int}).race.grep( &is-colorful ),
" - {($c / (exp($_,10) - exp($_-1,10) ) * 100).round(.001)}%";
$total += $c;
}
say "\nTotal colorful numbers: $total";</lang>
- Output:
Colorful numbers less than 100: 0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98 Largest magnitude colorful number: 98746253 Count of colorful numbers for each order of magnitude: 1 digit colorful number count: 10 - 100% 2 digit colorful number count: 56 - 62.222% 3 digit colorful number count: 328 - 36.444% 4 digit colorful number count: 1540 - 17.111% 5 digit colorful number count: 5514 - 6.127% 6 digit colorful number count: 13956 - 1.551% 7 digit colorful number count: 21596 - 0.24% 8 digit colorful number count: 14256 - 0.016% Total colorful numbers: 57256
Wren
<lang ecmascript>import "./math" for Int, Nums import "./set" for Set import "./seq" for Lst import "./fmt" for Fmt
var isColorful = Fn.new { |n|
if (n < 0) return false
if (n < 10) return true
var digits = Int.digits(n)
if (digits.contains(0) || digits.contains(1)) return false
var set = Set.new(digits)
var dc = digits.count
if (set.count < dc) return false
for (k in 2..dc) {
for (i in 0..dc-k) {
var prod = 1
for (j in i..i+k-1) prod = prod * digits[j]
if (set.contains(prod)) return false
set.add(prod)
}
}
return true
}
System.print("The colorful numbers less than 100 are:") var colorful = (0..99).where { |i| isColorful.call(i) }.toList for (chunk in Lst.chunks(colorful, 10)) Fmt.print("$2d", chunk)
var largest = 0 System.print("\nThe largest possible colorful number is:") for (i in 1e8-1..0) {
if (isColorful.call(i)) {
Fmt.print("$,d", i)
largest = i
break
}
}
var count = List.filled(9, 0) var dc = 1 var pow = 10 System.print("\nCount of colorful numbers for each order of magnitude:") var i = 0 while (true) {
if (dc > 1) {
var rem = i % 10
if (rem == 0 || rem == 1) {
i = i + 2 - rem
continue
}
}
if (isColorful.call(i)) count[dc] = count[dc] + 1
if (i == pow - 1 || i == largest) {
var total = (dc == 1) ? 10 : pow * 0.9
var pc = 100 * count[dc] / total
Fmt.print(" $d digit colorful number count: $,6d - $7.3f\%", dc, count[dc], pc)
if (i == largest) break
dc = dc + 1
pow = pow * 10
i = pow * 0.2 + 2
} else {
i = i + 1
}
}
Fmt.print("\nTotal colorful numbers: $,d", Nums.sum(count))</lang>
- Output:
The colorful numbers less than 100 are: 0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98 The largest possible colorful number is: 98,746,253 Count of colorful numbers for each order of magnitude: 1 digit colorful number count: 10 - 100.000% 2 digit colorful number count: 56 - 62.222% 3 digit colorful number count: 328 - 36.444% 4 digit colorful number count: 1,540 - 17.111% 5 digit colorful number count: 5,514 - 6.127% 6 digit colorful number count: 13,956 - 1.551% 7 digit colorful number count: 21,596 - 0.240% 8 digit colorful number count: 14,256 - 0.016% Total colorful numbers: 57,256
XPL0
<lang XPL0>func IPow(A, B); \A^B int A, B, T, I; [T:= 1; for I:= 1 to B do T:= T*A; return T; ];
func Colorful(N); \Return 'true' if N is a colorful number int N, Digits, R, I, J, Prod; def Size = 9*8*7*6*5*4*3*2 + 1; char Used(Size), Num(10); [if N < 10 then return true; \single digit number is colorful FillMem(Used, false, 10); \digits must be unique Digits:= 0; repeat N:= N/10; \slice digits off N
R:= rem(0);
if N=1 or R=0 or R=1 then return false;
if Used(R) then return false;
Used(R):= true; \digits must be unique
Num(Digits):= R;
Digits:= Digits+1;
until N = 0; FillMem(Used+10, false, Size-10); \products must be unique for I:= 0 to Digits-2 do
[Prod:= Num(I);
for J:= I+1 to Digits-1 do
[Prod:= Prod * Num(J);
if Used(Prod) then return false;
Used(Prod):= true;
];
];
return true; ];
int Count, N, Power, Total; [Text(0, "Colorful numbers less than 100: "); Count:= 0; for N:= 0 to 99 do
if Colorful(N) then
[IntOut(0, N);
Count:= Count+1;
if rem(Count/10) then ChOut(0, 9\tab\) else CrLf(0);
];
Text(0, "
Largest magnitude colorful number: "); N:= 98_765_432; loop [if Colorful(N) then quit;
N:= N-1;
];
IntOut(0, N); Text(0, "
Count of colorful numbers for each order of magnitude: "); Total:= 0; for Power:= 1 to 8 do
[Count:= if Power=1 then 1 else 0;
for N:= IPow(10, Power-1) to IPow(10, Power)-1 do
if Colorful(N) then Count:= Count+1;
IntOut(0, Power);
Text(0, " digit colorful number count: ");
IntOut(0, Count);
CrLf(0);
Total:= Total + Count;
];
Text(0, " Total colorful numbers: "); IntOut(0, Total); CrLf(0); ]</lang>
- Output:
Colorful numbers less than 100: 0 1 2 3 4 5 6 7 8 9 23 24 25 26 27 28 29 32 34 35 36 37 38 39 42 43 45 46 47 48 49 52 53 54 56 57 58 59 62 63 64 65 67 68 69 72 73 74 75 76 78 79 82 83 84 85 86 87 89 92 93 94 95 96 97 98 Largest magnitude colorful number: 98746253 Count of colorful numbers for each order of magnitude: 1 digit colorful number count: 10 2 digit colorful number count: 56 3 digit colorful number count: 328 4 digit colorful number count: 1540 5 digit colorful number count: 5514 6 digit colorful number count: 13956 7 digit colorful number count: 21596 8 digit colorful number count: 14256 Total colorful numbers: 57256