# Seven-sided dice from five-sided dice

Seven-sided dice from five-sided dice
You are encouraged to solve this task according to the task description, using any language you may know.

(Given an equal-probability generator of one of the integers 1 to 5 as `dice5`),   create `dice7` that generates a pseudo-random integer from 1 to 7 in equal probability using only `dice5` as a source of random numbers,   and check the distribution for at least one million calls using the function created in   Simple Random Distribution Checker.

Implementation suggestion: `dice7` might call `dice5` twice, re-call if four of the 25 combinations are given, otherwise split the other 21 combinations into 7 groups of three, and return the group index from the rolls.

## 11l

Translation of: Python
```F dice5()
R random:(1..5)

F dice7() -> Int
V r = dice5() + dice5() * 5 - 6
R I r < 21 {(r % 7) + 1} E dice7()

F distcheck(func, repeats, delta)
V bin = DefaultDict[Int, Int]()
L 1..repeats
bin[func()]++
V target = repeats I/ bin.len
V deltacount = Int(delta / 100.0 * target)
assert(all(bin.values().map(count -> abs(@target - count) < @deltacount)), ‘Bin distribution skewed from #. +/- #.: #.’.format(target, deltacount, sorted(bin.items()).map((key, count) -> (key, @target - count))))
print(bin)

distcheck(dice5, 1000000, 1)
distcheck(dice7, 1000000, 1)```
Output:
```DefaultDict([1 = 199586, 2 = 200094, 3 = 198933, 4 = 200824, 5 = 200563])
DefaultDict([1 = 142478, 2 = 142846, 3 = 143056, 4 = 142894, 5 = 143052, 6 = 143147, 7 = 142527])
```

The specification of a package Random_57:

```package Random_57 is

type Mod_7 is mod 7;

function Random7 return Mod_7;
-- a "fast" implementation, minimazing the calls to the Random5 generator
function Simple_Random7 return Mod_7;
-- a simple implementation

end Random_57;
```

Implementation of Random_57:

``` with Ada.Numerics.Discrete_Random;

package body Random_57 is
type M5 is mod 5;

Gen: Rand_5.Generator;
function Random7 return Mod_7 is
N: Natural;

begin
loop
N :=  Integer(Rand_5.Random(Gen))* 5 + Integer(Rand_5.Random(Gen));
-- N is uniformly distributed in 0 .. 24
if N < 21 then
return Mod_7(N/3);
else -- (N-21) is in 0 .. 3
N := (N-21) * 5 +  Integer(Rand_5.Random(Gen)); -- N is in 0 .. 19
if N < 14 then
return Mod_7(N / 2);
else -- (N-14) is in 0 .. 5
N := (N-14) * 5 +  Integer(Rand_5.Random(Gen)); -- N is in 0 .. 29
if N < 28 then
return Mod_7(N/4);
else -- (N-28) is in 0 .. 1
N := (N-28) * 5 + Integer(Rand_5.Random(Gen)); -- 0 .. 9
if N < 7 then
return Mod_7(N);
else -- (N-7) is in 0, 1, 2
N := (N-7)* 5 + Integer(Rand_5.Random(Gen)); -- 0 .. 14
if N < 14 then
return Mod_7(N/2);
else -- (N-14) is 0. This is not useful for us!
null;
end if;
end if;
end if;
end if;
end if;
end loop;

end Random7;

function Simple_Random7 return Mod_7 is
N: Natural :=
Integer(Rand_5.Random(Gen))* 5 + Integer(Rand_5.Random(Gen));
-- N is uniformly distributed in 0 .. 24
begin
while N > 20 loop
N :=  Integer(Rand_5.Random(Gen))* 5 + Integer(Rand_5.Random(Gen));
end loop; -- Now I <= 20
return Mod_7(N / 3);
end Simple_Random7;

begin
Rand_5.Reset(Gen);
end Random_57;
```

A main program, using the Random_57 package:

```with Ada.Text_IO, Random_57;

procedure R57 is

use Random_57;

type Fun is access function return Mod_7;

function Rand return Mod_7 renames Random_57.Random7;
-- change this to "... renames Random_57.Simple_Random;" if you like

procedure Test(Sample_Size: Positive; Rand: Fun; Precision: Float := 0.3) is

Counter: array(Mod_7) of Natural := (others => 0);
Expected: Natural := Sample_Size/7;
Small: Mod_7 := Mod_7'First;
Large: Mod_7 := Mod_7'First;

Result: Mod_7;
begin
for I in 1 .. Sample_Size loop
Result := Rand.all;
Counter(Result) := Counter(Result) + 1;
end loop;
for J in Mod_7 loop
if Counter(J) < Counter(Small) then Small := J; end if;
if Counter(J) > Counter(Large)  then Large := J;  end if;
end loop;
Ada.Text_IO.Put_Line(" Large Bin: " & Integer'Image(Counter(Large)));

if Float(Counter(Small)*7) * (1.0+Precision) < Float(Sample_Size) then
elsif Float(Counter(Large)*7) * (1.0-Precision) > Float(Sample_Size) then
else
end if;
end Test;

begin
Test(    10_000, Rand'Access, 0.08);
Test(   100_000, Rand'Access, 0.04);
Test( 1_000_000, Rand'Access, 0.02);
Test(10_000_000, Rand'Access, 0.01);
end R57;
```
Output:
```Sample Size:  10000
Bins: 1368 1404 1435 1491 1483 1440 1379
Small Bin: 1368
Large Bin:  1491
Passed

Sample Size:  100000
Bins: 14385 14110 14362 14404 14362 14206 14171
Small Bin: 14110
Large Bin:  14404
Passed

Sample Size:  1000000
Bins: 143765 142384 142958 142684 142799 142956 142454
Small Bin: 142384
Large Bin:  143765
Passed

Sample Size:  10000000
Bins: 1429266 1428214 1428753 1427032 1428418 1428699 1429618
Small Bin: 1427032
Large Bin:  1429618
Passed```

## ALGOL 68

Translation of: C

- note: This specimen retains the original C coding style.

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d

C's version using no multiplications, divisions, or mod operators:

```PROC dice5 = INT:
1 + ENTIER (5*random);

PROC mulby5 = (INT n)INT:
ABS (BIN n SHL 2) + n;

PROC dice7 = INT: (
INT d55 := 0;
INT m := 1;
WHILE
m := ABS ((2r1 AND BIN m) SHL 2) + ABS (BIN m SHR 1);  # repeats 4 - 2 - 1 #
d55 := mulby5(mulby5(d55)) + mulby5(dice5) + dice5 - 6;
# WHILE # d55 < m DO SKIP OD;

m := 1;
WHILE d55>0 DO
d55 +:= m;
m := ABS (BIN d55 AND 2r111); # modulas by 8 #
d55 := ABS (BIN d55 SHR 3)    # divide by 8 #
OD;
m
);

PROC distcheck = (PROC INT dice, INT count, upb)VOID: (
[upb]INT sum; FOR i TO UPB sum DO sum[i] := 0 OD;
FOR i TO count DO sum[dice]+:=1 OD;
FOR i TO UPB sum WHILE print(whole(sum[i],0)); i /= UPB sum DO print(", ") OD;
print(new line)
);

main:
(
distcheck(dice5, 1000000, 5);
distcheck(dice7, 1000000, 7)
)```
Output:
```200598, 199852, 199939, 200602, 199009
143529, 142688, 142816, 142747, 142958, 142802, 142460
```

## AutoHotkey

```dice5()
{  Random, v, 1, 5
Return, v
}

dice7()
{  Loop
{  v := 5 * dice5() + dice5() - 6
IfLess v, 21, Return, (v // 3) + 1
}
}
```
```Distribution check:

Total elements = 10000

Margin = 3% --> Lbound = 1386, Ubound = 1471

Bucket 1 contains 1450 elements.
Bucket 2 contains 1374 elements. Skewed.
Bucket 3 contains 1412 elements.
Bucket 4 contains 1465 elements.
Bucket 5 contains 1370 elements. Skewed.
Bucket 6 contains 1485 elements. Skewed.
Bucket 7 contains 1444 elements.```

## BBC BASIC

```      MAXRND = 7
FOR r% = 2 TO 5
check% = FNdistcheck(FNdice7, 10^r%, 0.1)
PRINT "Over "; 10^r% " runs dice7 ";
IF check% THEN
PRINT "failed distribution check with "; check% " bin(s) out of range"
ELSE
PRINT "passed distribution check"
ENDIF
NEXT
END

DEF FNdice7
LOCAL x% : x% = FNdice5 + 5*FNdice5
IF x%>26 THEN = FNdice7 ELSE = (x%+1) MOD 7 + 1

DEF FNdice5 = RND(5)

DEF FNdistcheck(RETURN func%, repet%, delta)
LOCAL i%, m%, r%, s%, bins%()
DIM bins%(MAXRND)
FOR i% = 1 TO repet%
r% = FN(^func%)
bins%(r%) += 1
IF r%>m% m% = r%
NEXT
FOR i% = 1 TO m%
IF bins%(i%)/(repet%/m%) > 1+delta s% += 1
IF bins%(i%)/(repet%/m%) < 1-delta s% += 1
NEXT
= s%
```
Output:
```Over 100 runs dice7 failed distribution check with 4 bin(s) out of range
Over 1000 runs dice7 failed distribution check with 2 bin(s) out of range
Over 10000 runs dice7 passed distribution check
Over 100000 runs dice7 passed distribution check
```

## C

```int rand5()
{
int r, rand_max = RAND_MAX - (RAND_MAX % 5);
while ((r = rand()) >= rand_max);
return r / (rand_max / 5) + 1;
}

int rand5_7()
{
int r;
while ((r = rand5() * 5 + rand5()) >= 27);
return r / 3 - 1;
}

int main()
{
printf(check(rand5, 5, 1000000, .05) ? "flat\n" : "not flat\n");
printf(check(rand7, 7, 1000000, .05) ? "flat\n" : "not flat\n");
return 0;
}
```
Output:
```flat
flat
```

## C#

Translation of: Java
```using System;

public class SevenSidedDice
{
Random random = new Random();

static void Main(string[] args)
{
SevenSidedDice sevenDice = new SevenSidedDice();
Console.WriteLine("Random number from 1 to 7: "+ sevenDice.seven());
}

int seven()
{
int v=21;
while(v>20)
v=five()+five()*5-6;
return 1+v%7;
}

int five()
{
return 1 + random.Next(5);
}
}
```

## C++

This solution tries to minimize calls to the underlying d5 by reusing information from earlier calls.

```template<typename F> class fivetoseven
{
public:
fivetoseven(F f): d5(f), rem(0), max(1) {}
int operator()();
private:
F d5;
int rem, max;
};

template<typename F>
int fivetoseven<F>::operator()()
{
while (rem/7 == max/7)
{
while (max < 7)
{
int rand5 = d5()-1;
max *= 5;
rem = 5*rem + rand5;
}

int groups = max / 7;
if (rem >= 7*groups)
{
rem -= 7*groups;
max -= 7*groups;
}
}

int result = rem % 7;
rem /= 7;
max /= 7;
return result+1;
}

int d5()
{
return 5.0*std::rand()/(RAND_MAX + 1.0) + 1;
}

fivetoseven<int(*)()> d7(d5);

int main()
{
srand(time(0));
test_distribution(d5, 1000000, 0.001);
test_distribution(d7, 1000000, 0.001);
}
```

## Clojure

Uses the verify function defined in Verify distribution uniformity/Naive#Clojure

```(def dice5 #(rand-int 5))

(defn dice7 []
(quot (->> dice5                     ; do the following to dice5
(repeatedly 2)            ; call it twice
(apply #(+ %1 (* 5 %2)))  ; d1 + 5*d2 => 0..24
#()                       ; wrap that up in a function
repeatedly                ; make infinite sequence of the above
(drop-while #(> % 20))    ; throw away anything > 20
first)                    ; grab first acceptable element
3))                            ; divide by three rounding down

(doseq [n [100 1000 10000] [num count okay?] (verify dice7 n)]
(println "Saw" num count "times:"
(if okay? "that's" "   not") "acceptable"))
```
```Saw 0 10 times:    not acceptable
Saw 1 19 times:    not acceptable
Saw 2 12 times:    not acceptable
Saw 3 15 times: that's acceptable
Saw 4 11 times:    not acceptable
Saw 5 11 times:    not acceptable
Saw 6 22 times:    not acceptable
Saw 0 142 times: that's acceptable
Saw 1 158 times:    not acceptable
Saw 2 151 times: that's acceptable
Saw 3 153 times: that's acceptable
Saw 4 118 times:    not acceptable
Saw 5 139 times: that's acceptable
Saw 6 139 times: that's acceptable
Saw 0 1498 times: that's acceptable
Saw 1 1411 times: that's acceptable
Saw 2 1436 times: that's acceptable
Saw 3 1434 times: that's acceptable
Saw 4 1414 times: that's acceptable
Saw 5 1408 times: that's acceptable
Saw 6 1399 times: that's acceptable```

## Common Lisp

Translation of: C
```(defun d5 ()
(1+ (random 5)))

(defun d7 ()
(loop for d55 = (+ (* 5 (d5)) (d5) -6)
until (< d55 21)
finally (return (1+ (mod d55 7)))))
```
```> (check-distribution 'd7 1000)
Distribution potentially skewed for 1: expected around 1000/7 got 153.
Distribution potentially skewed for 2: expected around 1000/7 got 119.
Distribution potentially skewed for 3: expected around 1000/7 got 125.
Distribution potentially skewed for 7: expected around 1000/7 got 156.
T
#<EQL Hash Table{7} 200B5A53>

> (check-distribution 'd7 10000)
NIL
#<EQL Hash Table{7} 200CB5BB>```

## D

Translation of: C++
```import std.random;
import verify_distribution_uniformity_naive: distCheck;

/// Generates a random number in [1, 5].
int dice5() /*pure nothrow*/ @safe {
return uniform(1, 6);
}

/// Naive, generates a random number in [1, 7] using dice5.
int fiveToSevenNaive() /*pure nothrow*/ @safe {
immutable int r = dice5() + dice5() * 5 - 6;
return (r < 21) ? (r % 7) + 1 : fiveToSevenNaive();
}

/**
Generates a random number in [1, 7] using dice5,
minimizing calls to dice5.
*/
int fiveToSevenSmart() @safe {
static int rem = 0, max = 1;

while (rem / 7 == max / 7) {
while (max < 7) {
immutable int rand5 = dice5() - 1;
max *= 5;
rem = 5 * rem + rand5;
}

immutable int groups = max / 7;
if (rem >= 7 * groups) {
rem -= 7 * groups;
max -= 7 * groups;
}
}

immutable int result = rem % 7;
rem /= 7;
max /= 7;
return result + 1;
}

void main() /*@safe*/ {
enum int N = 400_000;
distCheck(&dice5, N, 1);
distCheck(&fiveToSevenNaive, N, 1);
distCheck(&fiveToSevenSmart, N, 1);
}
```
Output:
```1 80365
2 79941
3 80065
4 79784
5 79845

1 57186
2 57201
3 57180
4 57231
5 57124
6 56832
7 57246

1 57367
2 56869
3 57644
4 57111
5 57157
6 56809
7 57043```

## E

Translation of: Common Lisp
 This example is in need of improvement: Write dice7 in a prettier fashion and use the distribution checker once it's been written.
```def dice5() {
return entropy.nextInt(5) + 1
}

def dice7() {
var d55 := null
while ((d55 := 5 * dice5() + dice5() - 6) >= 21) {}
return d55 %% 7 + 1
}```
```def bins := ([0] * 7).diverge()
for x in 1..1000 {
bins[dice7() - 1] += 1
}
println(bins.snapshot())```

## EasyLang

```func dice5 .
return randint 5
.
func dice25 .
return (dice5 - 1) * 5 + dice5
.
func dice7a .
return dice25 mod1 7
.
func dice7b .
repeat
h = dice25
until h <= 21
.
return h mod1 7
.
numfmt 3 0
n = 1000000
len dist[] 7
#
proc checkdist . .
for i to len dist[]
h = dist[i] / n * 7
if abs (h - 1) > 0.01
.
dist[i] = 0
print h
.
print "-> not uniform"
else
print "-> uniform"
.
.
#
for i to n
dist[dice7a] += 1
.
checkdist
#
print ""
for i to n
dist[dice7b] += 1
.
checkdist```
Output:
```1.122
1.118
1.121
1.117
0.840
0.842
0.840
-> not uniform

0.996
1.003
1.001
0.997
1.004
0.998
1.001
-> uniform
```

## Elixir

```defmodule Dice do
def dice5, do: :rand.uniform( 5 )

def dice7 do
dice7_from_dice5
end

defp dice7_from_dice5 do
d55 = 5*dice5 + dice5 - 6           # 0..24
if d55 < 21, do: rem( d55, 7 ) + 1,
else: dice7_from_dice5
end
end

fun5 = fn -> Dice.dice5 end
IO.inspect VerifyDistribution.naive( fun5, 1000000, 3 )
fun7 = fn -> Dice.dice7 end
IO.inspect VerifyDistribution.naive( fun7, 1000000, 3 )
```
Output:
```:ok
:ok
```

## Erlang

```-module( dice ).

dice5() -> random:uniform( 5 ).

dice7() ->
dice7_small_enough( dice5() * 5 + dice5() - 6 ). % 0 - 24

verify_distribution_uniformity:naive( fun dice7/0, 1000000, 1 ).

dice7_small_enough( N ) when N < 21 -> N div 3 + 1;
dice7_small_enough( _N ) -> dice7().
```
Output:
```76> dice:task().
ok
```

## Factor

```USING: kernel random sequences assocs locals sorting prettyprint
math math.functions math.statistics math.vectors math.ranges ;
IN: rosetta-code.dice7

! Output a random integer 1..5.
: dice5 ( -- x )
5 [1,b] random
;

! Output a random integer 1..7 using dice5 as randomness source.
: dice7 ( -- x )
0 [ dup 21 < ] [ drop dice5 5 * dice5 + 6 - ] do until
7 rem 1 +
;

! Roll the die by calling the quotation the given number of times and return
! an array with roll results.
! Sample call: 1000 [ dice7 ] roll
: roll ( times quot: ( -- x ) -- array )
[ call( --  x ) ] curry replicate
;

! Input array contains outcomes of a number of die throws. Each die result is
! an integer in the range 1..X. Calculate and return the number of each
! of the results in the array so that in the first position of the result
! there is the number of ones in the input array, in the second position
! of the result there is the number of twos in the input array, etc.
: count-dice-outcomes ( X array -- array )
histogram
swap [1,b] [ over [ 0 or ] change-at ] each
sort-keys values
;

! Verify distribution uniformity/Naive. Delta is the acceptable deviation
! from the ideal number of items in each bucket, expressed as a fraction of
! the total count. Sides is the number of die sides. Die-func is a word that
! produces a random number on stack in the range [1..sides], times is the
! number of times to call it.
! Sample call: 0.02 7 [ dice7 ] 100000 verify
:: verify ( delta sides die-func: ( -- random ) times -- )
sides
times die-func roll
count-dice-outcomes
dup .
times sides / :> ideal-count
ideal-count v-n vabs
times v/n
delta [ < ] curry all?
[ "Random enough" . ] [ "Not random enough" . ] if
;

! Call verify with 1, 10, 100, ... 1000000 rolls of 7-sided die.
: verify-all ( -- )
{ 1 10 100 1000 10000 100000 1000000 }
[| times | 0.02 7 [ dice7 ] times verify ] each
;
```
Output:
```USE: rosetta-code.dice7 verify-all
{ 0 0 0 1 0 0 0 }
"Not random enough"
{ 0 2 3 1 1 1 2 }
"Not random enough"
{ 17 12 15 11 13 13 19 }
"Not random enough"
{ 140 130 141 148 143 155 143 }
"Random enough"
{ 1457 1373 1427 1433 1443 1382 1485 }
"Random enough"
{ 14225 14320 14216 14326 14415 14084 14414 }
"Random enough"
{ 142599 141910 142524 143029 143353 142696 143889 }
"Random enough"```

## Forth

Works with: GNU Forth
```require random.fs

: d5 5 random 1+ ;
: discard? 5 = swap 1 > and ;
: d7
begin d5 d5 2dup discard? while 2drop repeat
1- 5 * + 1- 7 mod 1+ ;
```
Output:
```cr ' d7 1000000 7 1 check-distribution .
lower bound = 141429  upper bound = 144285
1 : 143241 ok
2 : 142397 ok
3 : 143522 ok
4 : 142909 ok
5 : 142001 ok
6 : 142844 ok
7 : 143086 ok
-1```

## Fortran

Works with: Fortran version 95 and later
```module rand_mod
implicit none

contains

function rand5()
integer :: rand5
real :: r

call random_number(r)
rand5 = 5*r + 1
end function

function rand7()
integer :: rand7

do
rand7 = 5*rand5() + rand5() - 6
if (rand7 < 21) then
rand7 = rand7 / 3 + 1
return
end if
end do
end function
end module

program Randtest
use rand_mod
implicit none

integer, parameter :: samples = 1000000

call distcheck(rand7, samples, 0.005)
write(*,*)
call distcheck(rand7, samples, 0.001)

end program
```
Output:
```Distribution Uniform

Distribution potentially skewed for bucket 1  Expected: 142857  Actual: 143142
Distribution potentially skewed for bucket 2  Expected: 142857  Actual: 143454
Distribution potentially skewed for bucket 3  Expected: 142857  Actual: 143540
Distribution potentially skewed for bucket 4  Expected: 142857  Actual: 142677
Distribution potentially skewed for bucket 5  Expected: 142857  Actual: 142511
Distribution potentially skewed for bucket 6  Expected: 142857  Actual: 142163
Distribution potentially skewed for bucket 7  Expected: 142857  Actual: 142513```

## FreeBASIC

Translation of: Liberty BASIC
```Function dice5() As Integer
Return Int(Rnd * 5) + 1
End Function

Function dice7() As Integer
Dim As Integer temp
Do
temp = dice5() * 5 + dice5() -6
Loop Until temp < 21
Return (temp Mod 7) +1
End Function

Dim Shared As Ulongint n = 1000000
Print "Testing "; n; " times"
If Not(distCheck(n, 0.05)) Then Print "Test failed" Else Print "Test passed"
Sleep
```
Output:
```Igual que la entrada de Liberty BASIC.
```

## Go

```package main

import (
"fmt"
"math"
"math/rand"
"time"
)

// "given"
func dice5() int {
return rand.Intn(5) + 1
}

// function specified by task "Seven-sided dice from five-sided dice"
func dice7() (i int) {
for {
i = 5*dice5() + dice5()
if i < 27 {
break
}
}
return (i / 3) - 1
}

// function specified by task "Verify distribution uniformity/Naive"
//
// Parameter "f" is expected to return a random integer in the range 1..n.
// (Values out of range will cause an unceremonious crash.)
// "Max" is returned as an "indication of distribution achieved."
// It is the maximum delta observed from the count representing a perfectly
// uniform distribution.
// Also returned is a boolean, true if "max" is less than threshold
// parameter "delta."
func distCheck(f func() int, n int,
repeats int, delta float64) (max float64, flatEnough bool) {
count := make([]int, n)
for i := 0; i < repeats; i++ {
count[f()-1]++
}
expected := float64(repeats) / float64(n)
for _, c := range count {
max = math.Max(max, math.Abs(float64(c)-expected))
}
return max, max < delta
}

// Driver, produces output satisfying both tasks.
func main() {
rand.Seed(time.Now().UnixNano())
const calls = 1000000
max, flatEnough := distCheck(dice7, 7, calls, 500)
fmt.Println("Max delta:", max, "Flat enough:", flatEnough)
max, flatEnough = distCheck(dice7, 7, calls, 500)
fmt.Println("Max delta:", max, "Flat enough:", flatEnough)
}
```
Output:
```Max delta: 356.1428571428696 Flat enough: true
Max delta: 787.8571428571304 Flat enough: false
```

## Groovy

```random = new Random()

int rand5() {
random.nextInt(5) + 1
}

int rand7From5() {
def raw = 25
while (raw > 21) {
raw = 5*(rand5() - 1) + rand5()
}
(raw % 7) + 1
}
```

Test:

```def test = {
(1..6). each {
def counts = [0g, 0g, 0g, 0g, 0g, 0g, 0g]
def target = 10g**it
def popSize = 7*target
(0..<(popSize)).each {
def i = rand7From5() - 1
counts[i] = counts[i] + 1g
}
BigDecimal stdDev = (counts.collect { it - target}.collect { it * it }.sum() / popSize) ** 0.5g
def countMap = (0..<counts.size()).inject([:]) { map, index -> map + [(index+1):counts[index]] }

println """\
counts: \${countMap}
population size: \${popSize}
std dev: \${stdDev.round(new java.math.MathContext(3))}
"""
}
}

4.times {
println """
TRIAL #\${it+1}
=============="""
test(it)
}
```
Output:
```TRIAL #1
==============
counts: [1:16, 2:10, 3:9, 4:7, 5:12, 6:8, 7:8]
population size: 70
std dev: 0.910

counts: [1:85, 2:97, 3:108, 4:110, 5:95, 6:105, 7:100]
population size: 700
std dev: 0.800

counts: [1:990, 2:1008, 3:992, 4:1060, 5:1008, 6:997, 7:945]
population size: 7000
std dev: 0.995

counts: [1:9976, 2:10007, 3:10009, 4:9858, 5:10109, 6:9988, 7:10053]
population size: 70000
std dev: 0.714

counts: [1:100310, 2:99783, 3:99843, 4:100353, 5:99804, 6:99553, 7:100354]
population size: 700000
std dev: 0.968

counts: [1:999320, 2:1000942, 3:1000201, 4:1000878, 5:999181, 6:999632, 7:999846]
population size: 7000000
std dev: 0.654

TRIAL #2
==============
counts: [1:10, 2:8, 3:9, 4:9, 5:14, 6:7, 7:13]
population size: 70
std dev: 0.756

counts: [1:104, 2:101, 3:97, 4:108, 5:100, 6:87, 7:103]
population size: 700
std dev: 0.619

counts: [1:995, 2:970, 3:1001, 4:953, 5:1006, 6:1081, 7:994]
population size: 7000
std dev: 1.18

counts: [1:10013, 2:10063, 3:9843, 4:9984, 5:9986, 6:10059, 7:10052]
population size: 70000
std dev: 0.711

counts: [1:100048, 2:99647, 3:100240, 4:100683, 5:99813, 6:100320, 7:99249]
population size: 700000
std dev: 1.39

counts: [1:1000579, 2:1000541, 3:999497, 4:1000805, 5:999708, 6:999161, 7:999709]
population size: 7000000
std dev: 0.586

TRIAL #3
==============
counts: [1:9, 2:8, 3:11, 4:14, 5:10, 6:11, 7:7]
population size: 70
std dev: 0.676

counts: [1:100, 2:92, 3:105, 4:107, 5:111, 6:91, 7:94]
population size: 700
std dev: 0.733

counts: [1:1010, 2:1053, 3:967, 4:981, 5:1027, 6:959, 7:1003]
population size: 7000
std dev: 0.984

counts: [1:9857, 2:10037, 3:9992, 4:10231, 5:9828, 6:10140, 7:9915]
population size: 70000
std dev: 1.37

counts: [1:99650, 2:99580, 3:99848, 4:100507, 5:99916, 6:100212, 7:100287]
population size: 700000
std dev: 1.01

counts: [1:1001710, 2:999667, 3:1000685, 4:1000411, 5:999369, 6:998469, 7:999689]
population size: 7000000
std dev: 0.965

TRIAL #4
==============
counts: [1:12, 2:7, 3:11, 4:12, 5:7, 6:9, 7:12]
population size: 70
std dev: 0.676

counts: [1:97, 2:96, 3:101, 4:93, 5:96, 6:124, 7:93]
population size: 700
std dev: 1.01

counts: [1:985, 2:1023, 3:1018, 4:1023, 5:995, 6:973, 7:983]
population size: 7000
std dev: 0.615

counts: [1:9948, 2:9968, 3:10131, 4:10050, 5:9990, 6:10039, 7:9874]
population size: 70000
std dev: 0.764

counts: [1:100125, 2:99616, 3:99912, 4:100286, 5:99674, 6:100190, 7:100197]
population size: 700000
std dev: 0.787

counts: [1:1001267, 2:999911, 3:1000602, 4:999483, 5:1000549, 6:998725, 7:999463]
population size: 7000000
std dev: 0.798```

```import System.Random
import Data.List

sevenFrom5Dice = do
d51 <- randomRIO(1,5) :: IO Int
d52 <- randomRIO(1,5) :: IO Int
let d7 = 5*d51+d52-6
if d7 > 20 then sevenFrom5Dice
else return \$ 1 + d7 `mod` 7
```
Output:
```*Main> replicateM 10 sevenFrom5Dice
[2,3,1,1,6,2,5,6,5,3]
```

Test:

```*Main> mapM_ print .sort =<< distribCheck sevenFrom5Dice 1000000 3
(1,(142759,True))
(2,(143078,True))
(3,(142706,True))
(4,(142403,True))
(5,(142896,True))
(6,(143028,True))
(7,(143130,True))
```

## Icon and Unicon

Translation of: Ruby

Uses `verify_uniform` from here.

```\$include "distribution-checker.icn"

# return a uniformly distributed number from 1 to 7,
# but only using a random number in range 1 to 5.
procedure die_7 ()
while rnd := 5*?5 + ?5 - 6 do {
if rnd < 21 then suspend rnd % 7 + 1
}
end

procedure main ()
if verify_uniform (create (|die_7()), 1000000, 0.01)
then write ("uniform")
else write ("skewed")
end
```
Output:
```5 142870
2 142812
7 142901
4 142960
1 143113
6 142706
3 142638
uniform
```

## J

The first step is to create 7-sided dice rolls from 5-sided dice rolls (`rollD5`):

```rollD5=: [: >: ] ?@\$ 5:      NB. makes a y shape array of 5s, "rolls" the array and increments.
roll2xD5=: [: rollD5 2 ,~ */ NB. rolls D5 twice for each desired D7 roll (y rows, 2 cols)
toBase10=: 5 #. <:           NB. decrements and converts rows from base 5 to 10
keepGood=: #~ 21&>           NB. compress out values not less than 21
groupin3s=: [: >. >: % 3:    NB. increments, divides by 3 and takes ceiling

getD7=: groupin3s@keepGood@toBase10@roll2xD5
```

Here are a couple of variations on the theme that achieve the same result:

```getD7b=: 0 8 -.~ 3 >.@%~ 5 #. [: <:@rollD5 2 ,~ ]
getD7c=: [: (#~ 7&>:) 3 >.@%~ [: 5&#.&.:<:@rollD5 ] , 2:
```

The trouble is that we probably don't have enough D7 rolls yet because we compressed out any double D5 rolls that evaluated to 21 or more. So we need to accumulate some more D7 rolls until we have enough. J has two types of verb definition - tacit (arguments not referenced) and explicit (more conventional function definitions) illustrated below:

Here's an explicit definition that accumulates rolls from `getD7`:

```rollD7x=: monad define
n=. */y                             NB. product of vector y is total number of D7 rolls required
rolls=. ''                          NB. initialize empty noun rolls
while. n > #rolls do.               NB. checks if if enough D7 rolls accumulated
rolls=. rolls, getD7 >. 0.75 * n  NB. calcs 3/4 of required rolls and accumulates getD7 rolls
end.
y \$ rolls                           NB. shape the result according to the vector y
)
```

Here's a tacit definition that does the same thing:

```getNumRolls=: [: >. 0.75 * */@[       NB. calc approx 3/4 of the required rolls
accumD7Rolls=: ] , getD7@getNumRolls  NB. accumulates getD7 rolls
isNotEnough=: */@[ > #@]              NB. checks if enough D7 rolls accumulated

rollD7t=: ] \$ (accumD7Rolls ^: isNotEnough ^:_)&''
```

The `verb1 ^: verb2 ^:_` construct repeats `x verb1 y` while `x verb2 y` is true. It is like saying "Repeat accumD7Rolls while isNotEnough".

Example usage:

```   rollD7t 10         NB. 10 rolls of D7
6 4 5 1 4 2 4 5 2 5
rollD7t 2 5        NB. 2 by 5 array of D7 rolls
5 1 5 1 3
3 4 3 5 6
rollD7t 2 3 5      NB. 2 by 3 by 5 array of D7 rolls
4 7 7 5 7
3 7 1 4 5
5 4 5 7 6

1 1 7 6 3
4 4 1 4 4
1 1 1 6 5

NB. check results from rollD7x and rollD7t have same shape
(\$@rollD7x -: \$@rollD7t) 10
1
(\$@rollD7x -: \$@rollD7t) 2 3 5
1
```

## Java

Translation of: Python
```import java.util.Random;
public class SevenSidedDice
{
private static final Random rnd = new Random();
public static void main(String[] args)
{
SevenSidedDice now=new SevenSidedDice();
System.out.println("Random number from 1 to 7: "+now.seven());
}
int seven()
{
int v=21;
while(v>20)
v=five()+five()*5-6;
return 1+v%7;
}
int five()
{
return 1+rnd.nextInt(5);
}
}
```

## JavaScript

Translation of: Ruby
```function dice5()
{
return 1 + Math.floor(5 * Math.random());
}

function dice7()
{
while (true)
{
var dice55 = 5 * dice5() + dice5() - 6;
if (dice55 < 21)
return dice55 % 7 + 1;
}
}

distcheck(dice5, 1000000);
print();
distcheck(dice7, 1000000);
```
Output:
```1       199792
2       200425
3       199243
4       200407
5       200133

1       143617
2       142209
3       143023
4       142990
5       142894
6       142648
7       142619 ```

## jq

Works with: jq

Also works with gojq, the Go implementation of jq.

In this entry, the results for both a low-entropy and a high-entropy 5-sided die are shown. The former uses the computer's clock as a not-very-good PRNG, and the latter uses /dev/random in accordance with the following invocation:

```#!/bin/bash
< /dev/random tr -cd '0-9' | fold -w 1 | jq -Mcnr -f dice.jq
```

The results employ a two-tailed χ2-test at the 95% confidence level according to which we are entitled to reject the null hypothesis of uniform randomness if the χ2 statistic is less than 1.69 or greater than 16.013, assuming the number of trials is large enough. See https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm

dice.jq

```# Output: a PRN in range(0;\$n) where \$n is .
def prn:
if . == 1 then 0
else . as \$n
| ((\$n-1)|tostring|length) as \$w
| [limit(\$w; inputs)] | join("") | tonumber
| if . < \$n then . else (\$n | prn) end
end;

# Emit a stream of [value, frequency] pairs
def histogram(stream):
reduce stream as \$s ({};
(\$s|type) as \$t
| (if \$t == "string" then \$s else (\$s|tojson) end) as \$y
| .[\$t][\$y][0] = \$s
| .[\$t][\$y][1] += 1 )
| .[][] ;

# sum of squares
def ss(s): reduce s as \$x (0; . + (\$x * \$x));

def chiSquared(\$expected):
debug # show the actual frequencies
| ss( .[] - \$expected ) / \$expected;

# The high-entropy 5-sided die
def dice5: 1 + (5|prn);

# The low-entropy 5-sided die
def pseudo_dice5:
def r: (now * 100000 | floor) % 10;
null | until(. and (. < 5); r) | 1 + . ;

# The formal argument dice5 should behave like a 5-sided dice:
def dice7(dice5):
1 +  ([limit(7; repeat(dice5))]|add % 7) ;

# Issue a report on the results of a sequence of \$n trials using the specified dice
def report(dice; \$n):
1.69 as \$lower
| 16.013 as \$upper
| [histogram( limit(\$n; repeat(dice)) ) | last]
| chiSquared(\$n/7) as \$x2
| "The χ2 statistic for a trial of \(\$n) virtual tosses is \(\$x2).",
"Using a two-sided χ2-test with seven degrees of freedom (\(\$lower), \(\$upper)), it is reasonable to conclude that",
(if   \$x2 < \$lower then "this is lower than would be expected for a fair die."
elif \$x2 > \$upper then "this is higher than would be expected for a fair die."
else "this is consistent with the die being fair."
end) ;

def report(\$n):
"Low-entropy die results:",
report(dice7(pseudo_dice5); \$n),
"",
"High-entropy die results:",
report(dice7(dice5); \$n) ;

report(70)```
Output:
```Low-entropy die results:
["DEBUG:",[19,14,6,18,7,5,1]]
The χ2 statistic for a trial of 70 virtual tosses is 29.2.
Using a two-sided χ2-test with seven degrees of freedom (1.69, 16.013), it is reasonable to conclude that
this is higher than would be expected for a fair die.

High-entropy die results:
["DEBUG:",[9,11,9,10,15,11,5]]
The χ2 statistic for a trial of 70 virtual tosses is 5.4.
Using a two-sided χ2-test with seven degrees of freedom (1.69, 16.013), it is reasonable to conclude that
this is consistent with the die being fair.
```

Results for 1,000,000 trials:

```Low-entropy die results:
["DEBUG:",[41440,57949,15946,44821,117168,339337,383339]]
The χ2 statistic for a trial of 1000000 virtual tosses is 982157.0011039999.
Using a two-sided χ2-test with seven degrees of freedom (1.69, 16.013), it is reasonable to conclude that
this is higher than would be expected for a fair die.

High-entropy die results:
["DEBUG:",[142860,143087,142213,142065,143359,143494,142922]]
The χ2 statistic for a trial of 1000000 virtual tosses is 12.298347999999999.
Using a two-sided χ2-test with seven degrees of freedom (1.69, 16.013), it is reasonable to conclude that
this is consistent with the die being fair.
```

## Julia

```using Random: seed!
seed!(1234) # for reproducibility

dice5() = rand(1:5)

function dice7()
while true
a = dice5()
b = dice5()
c = a + 5(b - 1)
if c <= 21
return mod1(c, 7)
end
end
end

rolls = (dice7() for i in 1:100000)
roll_counts = Dict{Int,Int}()
for roll in rolls
roll_counts[roll] = get(roll_counts, roll, 0) + 1
end
foreach(println, sort(roll_counts))
```

Output:

```1 => 14530
2 => 13872
3 => 14422
4 => 14425
5 => 14323
6 => 14315
7 => 14113
```

## Kotlin

```// version 1.1.3

import java.util.Random

val r = Random()

fun dice5() = 1 + r.nextInt(5)

fun dice7(): Int {
while (true) {
val t = (dice5() - 1) * 5 + dice5() - 1
if (t >= 21) continue
return 1 + t / 3
}
}

fun checkDist(gen: () -> Int, nRepeats: Int, tolerance: Double = 0.5) {
val occurs = mutableMapOf<Int, Int>()
for (i in 1..nRepeats) {
val d = gen()
if (occurs.containsKey(d))
occurs[d] = occurs[d]!! + 1
else
occurs.put(d, 1)
}
val expected = (nRepeats.toDouble()/ occurs.size).toInt()
val maxError = (expected * tolerance / 100.0).toInt()
println("Repetitions = \$nRepeats, Expected = \$expected")
println("Tolerance = \$tolerance%, Max Error = \$maxError\n")
println("Integer   Occurrences   Error  Acceptable")
val f = "  %d        %5d      %5d     %s"
var allAcceptable = true
for ((k,v) in occurs.toSortedMap()) {
val error = Math.abs(v - expected)
val acceptable = if (error <= maxError) "Yes" else "No"
if (acceptable == "No") allAcceptable = false
println(f.format(k, v, error, acceptable))
}
println("\nAcceptable overall: \${if (allAcceptable) "Yes" else "No"}")
}

fun main(args: Array<String>) {
checkDist(::dice7, 1_400_000)
}
```

Sample output:

```Repetitions = 1400000, Expected = 200000
Tolerance = 0.5%, Max Error = 1000

Integer   Occurrences   Error  Acceptable
1        199285        715     Yes
2        200247        247     Yes
3        199709        291     Yes
4        199983         17     Yes
5        199990         10     Yes
6        200664        664     Yes
7        200122        122     Yes

Acceptable overall: Yes
```

## Liberty BASIC

```n=1000000    '1000000 would take several minutes
print "Testing ";n;" times"
if  not(check(n, 0.05)) then print "Test failed" else print "Test passed"
end

'function check(n, delta) is defined at
'http://rosettacode.org/wiki/Verify_distribution_uniformity/Naive#Liberty_BASIC

function GENERATOR()
'GENERATOR = int(rnd(0)*10) '0..9
'GENERATOR = 1+int(rnd(0)*5) '1..5: dice5

'dice7()
do
temp =dice5() *5 +dice5() -6
loop until temp <21
GENERATOR =( temp mod 7) +1

end function

function dice5()
dice5=1+int(rnd(0)*5) '1..5: dice5
end function```
Output:
```Testing 1000000 times
minVal        Expected      maxVal
135714        142857        150000
Bucket        Counter       pass/fail
1             143310
2             143500
3             143040
4             145185
5             140998
6             142610
7             141357
Test passed
```

## Lua

```dice5 = function() return math.random(5) end

function dice7()
x = dice5() * 5 + dice5() - 6
if x > 20 then return dice7() end
return x%7 + 1
end
```

## M2000 Interpreter

We make a stack object (is reference type) and pass it as a closure to dice7 lambda function. For each dice7 we pop the top value of stack, and we add a fresh dice5 (random(1,5)) as last value of stack, so stack used as FIFO. Each time z has the sum of 7 random values.

We check for uniform numbers using +-5% from expected value.

```Module CheckIt {
Def long i, calls, max, min
s=stack:=random(1,5),random(1,5), random(1,5), random(1,5), random(1,5), random(1,5), random(1,5)
z=0: for i=1 to 7 { z+=stackitem(s, i)}
dice7=lambda z, s -> {
=((z-1) mod 7)+1 : stack s {z-=Number : data random(1,5): z+=Stackitem(7)}
}
Dim count(1 to 7)=0&  ' long type
calls=700000
p=0.05
IsUniform=lambda max=calls/7*(1+p), min=calls/7*(1-p) (a)->{
if len(a)=0 then =false : exit
=false
m=each(a)
while m
if array(m)<min or array(m)>max then break
end while
=true
}
For i=1 to calls {count(dice7())++}
max=count()#max()
expected=calls div 7
min=count()#min()
for i=1 to 7
document doc\$=format\$("{0}{1::-7}",i,count(i))+{
}
Next i
doc\$=format\$("min={0} expected={1} max={2}", min, expected, max)+{
}+format\$("Verify Uniform:{0}", if\$(IsUniform(count())->"uniform", "skewed"))+{
}
Print
report doc\$
clipboard doc\$
}
CheckIt```
Output:
```1   9865
2  10109
3   9868
4   9961
5   9936
6   9922
7  10339
min=9865 expected=10000 max=10339
Verify Uniform:uniform

1 100214
2 100336
3 100049
4  99505
5  99951
6  99729
7 100216
min=99505 expected=100000 max=100336
Verify Uniform:uniform
```

## Mathematica /Wolfram Language

```sevenFrom5Dice := (tmp\$ = 5*RandomInteger[{1, 5}] + RandomInteger[{1, 5}] - 6;
If [tmp\$ < 21, 1 + Mod[tmp\$ , 7], sevenFrom5Dice])
```
```CheckDistribution[sevenFrom5Dice, 1000000, 5]
->Expected: 142857., Generated :{142206,142590,142650,142693,142730,143475,143656}
->"Flat"```

## Nim

We use the distribution checker from task Simple Random Distribution Checker.

```import random, tables

proc dice5(): int = rand(1..5)

proc dice7(): int =
while true:
let val = 5 * dice5() + dice5() - 6
if val < 21:
return val div 3 + 1

proc checkDist(f: proc(): int; repeat: Positive; tolerance: float) =

var counts: CountTable[int]
for _ in 1..repeat:
counts.inc f()

let expected = (repeat / counts.len).toInt    # Rounded to nearest.
let allowedDelta = (expected.toFloat * tolerance / 100).toInt
var maxDelta = 0
for val, count in counts.pairs:
let d = abs(count - expected)
if d > maxDelta: maxDelta = d

let status = if maxDelta <= allowedDelta: "passed" else: "failed"
echo "Checking ", repeat, " values with a tolerance of ", tolerance, "%."
echo "Random generator ", status, " the uniformity test."
echo "Max delta encountered = ", maxDelta, "   Allowed delta = ", allowedDelta

when isMainModule:
import random
randomize()
checkDist(dice7, 1_000_000, 0.5)
```
Output:
```Checking 1000000 values with a tolerance of 0.5%.
Random generator passed the uniformity test.
Max delta encountered = 552   Allowed delta = 714```

## OCaml

```let dice5() = 1 + Random.int 5 ;;

let dice7 =
let rolls2answer = Hashtbl.create 25 in
let n = ref 0 in
for roll1 = 1 to 5 do
for roll2 = 1 to 5 do
incr n
done;
done;
let rec aux() =
let trial = Hashtbl.find rolls2answer (dice5(),dice5()) in
if trial <= 7 then trial else aux()
in
aux
;;
```

## PARI/GP

```dice5()=random(5)+1;

dice7()={
my(t);
while((t=dice5()*5+dice5()) > 21,);
(t+2)\3
};```

## Pascal

A console application in Free Pascal, created with the Lazarus IDE.

The algorithm suggested in the task description requires on average 50/21 (about 2.38) calls to Dice5 for each call to Dice7. See the link in the VBA solution for a discussion on how to reduce this ratio. It cannot be made less than log_5(7) = 1.209062. The algorithm below is based on Rex Kerr's solution, and requires about 1.2185 calls to Dice5 per call to Dice7. Runtime is about 60% of that for the suggested simple algorithm.

A chi-squared test can be carried out with the help of statistical tables, and is preferred here to an arbitrary "naive" test.

```unit UConverter;
(*
Defines a converter object to output uniformly distributed random integers 1..7,
given a source of uniformly distributed random integers 1..5.
*)
interface

type
TFace5 = 1..5;
TFace7 = 1..7;
TDice5 = function() : TFace5;

type TConverter = class( TObject)
private
fDigitBuf: array [0..19] of integer; // holds digits in base 7
fBufCount, fBufPtr : integer;
fDice5 : TDice5;  // passed-in generator for integers 1..5
fNrDice5 : int64; // diagnostics, counts calls to fDice5
public
procedure Reset();
function Dice7() : TFace7;
property NrDice5 : int64 read fNrDice5;
end;

implementation

begin
inherited Create();
self.Reset();
end;

procedure TConverter.Reset();
begin
fBufCount := 0;
fBufPtr := 0;
fNrDice5 := 0;
end;

function TConverter.Dice7() : TFace7;
var
digit_holder, temp : int64;
j : integer;
begin
if fBufPtr = fBufCount then begin // if no more in buffer
fBufCount := 0;
fBufPtr := 0;
repeat // first time through will usually be enough
// Use supplied fDice5 to generate random 23-digit integer in base 5.
digit_holder := 0;
for j := 0 to 22 do begin
digit_holder := 5*digit_holder + fDice5() - 1;
inc( fNrDice5);
end;
// Convert to 20-digit number in base 7. (A simultaneous DivMod
// procedure would be neater, but isn't available for int64.)
for j := 0 to 19 do begin
temp := digit_holder div 7;
fDigitBuf[j] := digit_holder - 7*temp;
digit_holder := temp;
end;
// Maximum possible is 5^23 - 1, which is 10214646460315315132 in base 7.
// If leading digit in base 7 is 0 then low 19 digits are random.
// Else number begins with 100, 101, or 102; and if with
//   100 or 101 then low 17 digits are random. And so on.
if fDigitBuf[19] = 0 then fBufCount := 19
else if fDigitBuf[17] < 2 then fBufCount := 17
else if fDigitBuf[16] = 0 then fBufCount := 16;
// We could go on but that will do.
until fBufCount > 0;
end; // if no more in buffer
result := fDigitBuf[fBufPtr] + 1;
inc( fBufPtr);
end;
end.

program Dice_SevenFromFive;
(*
Demonstrates use of the UConverter unit.
*)
{\$mode objfpc}{\$H+}
uses
SysUtils, UConverter;

function Dice5() : UConverter.TFace5;
begin
result := Random(5) + 1; // Random(5) returns 0..4
end;

// Percentage points of the chi-squared distribution, 6 degrees of freedom.
// From New Cambridge Statistical Tables, 2nd edn, pp. 40-41.
const
CHI_SQ_6df_95pc = 1.635;
CHI_SQ_6df_05pc = 12.59;

// Main routine
var
nrThrows, j, k : integer;
nrFaces : array [1..7] of integer;
X2, expected, diff : double;
conv : UConverter.TConverter;
begin
conv := UConverter.TConverter.Create( @Dice5);
WriteLn( 'Enter 0 throws to quit');
repeat
WriteLn(''); Write( 'Number of throws (0 to quit): ');
if nrThrows = 0 then begin
conv.Free();
exit;
end;
conv.Reset(); // clears count of calls to Dice5
for k := 1 to 7 do nrFaces[k] := 0;
for j := 1 to nrThrows do begin
k := conv.Dice7();
inc( nrFaces[k]);
end;
WriteLn('');
WriteLn( SysUtils.Format( 'Number of throws = %10d', [nrThrows]));
WriteLn( SysUtils.Format( 'Calls to Dice5   = %10d', [conv.NrDice5]));
for k := 1 to 7 do
WriteLn( SysUtils.Format( '   Number of %d''s = %10d', [k, nrFaces[k]]));

// Calculation of chi-squared
expected := nrThrows/7.0;
X2 := 0.0;
for k := 1 to 7 do begin
diff := nrFaces[k] - expected;
X2 := X2 + diff*diff/expected;
end;
WriteLn( SysUtils.Format( 'X^2 = %0.3f on 6 degrees of freedom', [X2]));
if X2 < CHI_SQ_6df_95pc then      WriteLn( 'Too regular at 5% level')
else if X2 > CHI_SQ_6df_05pc then WriteLn( 'Too irregular at 5% level')
else                              WriteLn( 'Satisfactory at 5% level')
until false;
end.
```
Output:
```Number of throws =  100000000
Calls to Dice5   =  121846341
Number of 1's =   14282807
Number of 2's =   14282277
Number of 3's =   14288393
Number of 4's =   14285486
Number of 5's =   14289379
Number of 6's =   14291053
Number of 7's =   14280605
X^2 = 6.687 on 6 degrees of freedom
Satisfactory at 5% level
```

## Perl

Using dice5 twice to generate numbers in the range 0 to 24. If we consider these modulo 8 and re-call if we get zero, we have eliminated 4 cases and created the necessary number in the range from 1 to 7.

```sub dice5 { 1+int rand(5) }

sub dice7 {
while(1) {
my \$d7 = (5*dice5()+dice5()-6) % 8;
return \$d7 if \$d7;
}
}

my %count7;
my \$n = 1000000;
\$count7{dice7()}++ for 1..\$n;
printf "%s: %5.2f%%\n", \$_, 100*(\$count7{\$_}/\$n*7-1) for sort keys %count7;
```
Output:
```1:  0.05%
2:  0.16%
3: -0.43%
4:  0.11%
5:  0.01%
6: -0.15%
7:  0.24%
```

## Phix

replace rand7() in Verify_distribution_uniformity/Naive#Phix with:

```function dice5()
return rand(5)
end function

function dice7()
while true do
integer r = dice5()*5+dice5()-3         --  ( ie 3..27, but )
if r<24 then return floor(r/3) end if   -- (only 3..23 useful)
end while
end function
```
Output:
```1000000 iterations: flat
```

## PicoLisp

```(de dice5 ()
(rand 1 5) )

(de dice7 ()
(use R
(until (> 21 (setq R (+ (* 5 (dice5)) (dice5) -6))))
(inc (% R 7)) ) )```
Output:
```: (let R NIL
(do 1000000 (accu 'R (dice7) 1))
(sort R) )
-> ((1 . 142295) (2 . 142491) (3 . 143448) (4 . 143129) (5 . 142701) (6 . 143142) (7 . 142794))```

## PureBasic

Translation of: Lua
```Procedure dice5()
ProcedureReturn Random(4) + 1
EndProcedure

Procedure dice7()
Protected x

x = dice5() * 5 + dice5() - 6
If x > 20
ProcedureReturn dice7()
EndIf

ProcedureReturn x % 7 + 1
EndProcedure
```

## Python

```from random import randint

def dice5():
return randint(1, 5)

def dice7():
r = dice5() + dice5() * 5 - 6
return (r % 7) + 1 if r < 21 else dice7()
```

Distribution check using Simple Random Distribution Checker:

```>>> distcheck(dice5, 1000000, 1)
{1: 200244, 2: 199831, 3: 199548, 4: 199853, 5: 200524}
>>> distcheck(dice7, 1000000, 1)
{1: 142853, 2: 142576, 3: 143067, 4: 142149, 5: 143189, 6: 143285, 7: 142881}
```

## Quackery

```  [ 5 random 1+ ]   is dice5 ( --> n )

[ dice5 5 *
dice5 + 6 -
[ table
0 0 0 0 1
1 1 2 2 2
3 3 3 4 4
4 5 5 5 6
6 6 7 7 7 ]
dup 0 = iff
drop again ] is dice7 ( --> n )```
Output:

`distribution` is defined at Verify distribution uniformity/Naive#Quackery.

```/O> ' dice7 1000000 666 distribution
...
[ 143196 142815 143451 142716 142964 142300 142558 ]

Stack empty.```

## R

5-sided die.

```dice5 <- function(n=1) sample(5, n, replace=TRUE)
```

Simple but slow 7-sided die, using a while loop.

```dice7.while <- function(n=1)
{
score <- numeric()
while(length(score) < n)
{
total <- sum(c(5,1) * dice5(2)) - 3
if(total < 24) score <- c(score, total %/% 3)
}
score
}
system.time(dice7.while(1e6)) # longer than 4 minutes
```

More complex, but much faster vectorised version.

```dice7.vec <- function(n=1, checkLength=TRUE)
{
morethan2n <- 3 * n + 10 + (n %% 2)       #need more than 2*n samples, because some are discarded
twoDfive <- matrix(dice5(morethan2n), nrow=2)
total <- colSums(c(5, 1) * twoDfive) - 3
score <- ifelse(total < 24, total %/% 3, NA)
score <- score[!is.na(score)]
#If length is less than n (very unlikely), add some more samples
if(checkLength)
{
while(length(score) < n)
{
score <- c(score, dice7(n, FALSE))
}
score[1:n]
} else score
}
system.time(dice7.vec(1e6))   # ~1 sec
```

## Racket

```#lang racket

(define (dice7)
(define res (+ (* 5 (dice5)) (dice5) -6))
(if (< res 21) (+ 1 (modulo res 7)) (dice7)))
```

Checking the uniformity using math library:

```-> (require math/statistics)
-> (samples->hash (for/list ([i 700000]) (dice7)))
'#hash((7 . 100392)
(6 . 100285)
(5 . 99774)
(4 . 100000)
(3 . 100000)
(2 . 99927)
(1 . 99622))
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2018.03
```my \$d5 = 1..5;
sub d5() { \$d5.roll; }  # 1d5

sub d7() {
my \$flat = 21;
\$flat = 5 * d5() - d5() until \$flat < 21;
\$flat % 7 + 1;
}

# Testing
my @dist;
my \$n = 1_000_000;
my \$expect = \$n / 7;

loop (\$_ = \$n; \$n; --\$n) { @dist[d7()]++; }

say "Expect\t",\$expect.fmt("%.3f");
for @dist.kv -> \$i, \$v {
say "\$i\t\$v\t" ~ ((\$v - \$expect)/\$expect*100).fmt("%+.2f%%") if \$v;
}
```
Output:
```Expect	142857.143
1	143088	+0.16%
2	143598	+0.52%
3	141741	-0.78%
4	142832	-0.02%
5	143040	+0.13%
6	142988	+0.09%
7	142713	-0.10%
```

## REXX

```/*REXX program simulates a 7─sided die based on a 5─sided throw for a number of trials. */
parse arg trials sample seed .                   /*obtain optional arguments from the CL*/
if trials=='' | trials=","  then trials=       1 /*Not specified?  Then use the default.*/
if sample=='' | sample=","  then sample= 1000000 /* "      "         "   "   "     "    */
if datatype(seed, 'W')  then call random ,,seed  /*Integer?  Then use it as a RAND seed.*/
L= length(trials)                                /* [↑]  one million samples to be used.*/

do #=1  for trials;            die.= 0        /*performs the number of desired trials*/
k= 0
do  until k==sample;           r= 5 * random(1, 5)  +  random(1, 5)  -  6
if r>20  then iterate
k= k + 1;                      r= r // 7  +  1;          die.r= die.r + 1
end   /*until*/
say
expect= sample % 7
say center('trial:' right(#, L)    "   "     sample  'samples, expect' expect, 80, "─")

do j=1  for 7
say '      side'      j       "had "       die.j           ' occurrences',
'      difference from expected:'right(die.j - expect, length(sample) )
end   /*j*/
end   /*#*/                                   /*stick a fork in it,  we're all done. */
```
output   when using the input of:     11

(Shown at five-sixth size.)

```──────────────────trial:  1     1000000 samples, expect 142857──────────────────
side 1 had  142076  occurrences       difference from expected:   -781
side 2 had  143053  occurrences       difference from expected:    196
side 3 had  142342  occurrences       difference from expected:   -515
side 4 had  142633  occurrences       difference from expected:   -224
side 5 had  143024  occurrences       difference from expected:    167
side 6 had  143827  occurrences       difference from expected:    970
side 7 had  143045  occurrences       difference from expected:    188

──────────────────trial:  2     1000000 samples, expect 142857──────────────────
side 1 had  143470  occurrences       difference from expected:    613
side 2 had  142998  occurrences       difference from expected:    141
side 3 had  142654  occurrences       difference from expected:   -203
side 4 had  142545  occurrences       difference from expected:   -312
side 5 had  142452  occurrences       difference from expected:   -405
side 6 had  143144  occurrences       difference from expected:    287
side 7 had  142737  occurrences       difference from expected:   -120

──────────────────trial:  3     1000000 samples, expect 142857──────────────────
side 1 had  142773  occurrences       difference from expected:    -84
side 2 had  143198  occurrences       difference from expected:    341
side 3 had  142296  occurrences       difference from expected:   -561
side 4 had  142804  occurrences       difference from expected:    -53
side 5 had  142897  occurrences       difference from expected:     40
side 6 had  142382  occurrences       difference from expected:   -475
side 7 had  143650  occurrences       difference from expected:    793

──────────────────trial:  4     1000000 samples, expect 142857──────────────────
side 1 had  143150  occurrences       difference from expected:    293
side 2 had  142635  occurrences       difference from expected:   -222
side 3 had  142763  occurrences       difference from expected:    -94
side 4 had  142853  occurrences       difference from expected:     -4
side 5 had  143132  occurrences       difference from expected:    275
side 6 had  142403  occurrences       difference from expected:   -454
side 7 had  143064  occurrences       difference from expected:    207

──────────────────trial:  5     1000000 samples, expect 142857──────────────────
side 1 had  143041  occurrences       difference from expected:    184
side 2 had  142701  occurrences       difference from expected:   -156
side 3 had  143416  occurrences       difference from expected:    559
side 4 had  142097  occurrences       difference from expected:   -760
side 5 had  142451  occurrences       difference from expected:   -406
side 6 had  143332  occurrences       difference from expected:    475
side 7 had  142962  occurrences       difference from expected:    105

──────────────────trial:  6     1000000 samples, expect 142857──────────────────
side 1 had  142502  occurrences       difference from expected:   -355
side 2 had  142429  occurrences       difference from expected:   -428
side 3 had  143146  occurrences       difference from expected:    289
side 4 had  142791  occurrences       difference from expected:    -66
side 5 had  143271  occurrences       difference from expected:    414
side 6 had  143415  occurrences       difference from expected:    558
side 7 had  142446  occurrences       difference from expected:   -411

──────────────────trial:  7     1000000 samples, expect 142857──────────────────
side 1 had  142700  occurrences       difference from expected:   -157
side 2 had  142691  occurrences       difference from expected:   -166
side 3 had  143067  occurrences       difference from expected:    210
side 4 had  141562  occurrences       difference from expected:  -1295
side 5 had  143316  occurrences       difference from expected:    459
side 6 had  143150  occurrences       difference from expected:    293
side 7 had  143514  occurrences       difference from expected:    657

──────────────────trial:  8     1000000 samples, expect 142857──────────────────
side 1 had  142362  occurrences       difference from expected:   -495
side 2 had  143298  occurrences       difference from expected:    441
side 3 had  142639  occurrences       difference from expected:   -218
side 4 had  142811  occurrences       difference from expected:    -46
side 5 had  143275  occurrences       difference from expected:    418
side 6 had  142765  occurrences       difference from expected:    -92
side 7 had  142850  occurrences       difference from expected:     -7

──────────────────trial:  9     1000000 samples, expect 142857──────────────────
side 1 had  143508  occurrences       difference from expected:    651
side 2 had  142650  occurrences       difference from expected:   -207
side 3 had  142614  occurrences       difference from expected:   -243
side 4 had  142916  occurrences       difference from expected:     59
side 5 had  142944  occurrences       difference from expected:     87
side 6 had  143129  occurrences       difference from expected:    272
side 7 had  142239  occurrences       difference from expected:   -618

──────────────────trial: 10     1000000 samples, expect 142857──────────────────
side 1 had  142455  occurrences       difference from expected:   -402
side 2 had  143112  occurrences       difference from expected:    255
side 3 had  143435  occurrences       difference from expected:    578
side 4 had  142704  occurrences       difference from expected:   -153
side 5 had  142376  occurrences       difference from expected:   -481
side 6 had  142721  occurrences       difference from expected:   -136
side 7 had  143197  occurrences       difference from expected:    340

──────────────────trial: 11     1000000 samples, expect 142857──────────────────
side 1 had  142967  occurrences       difference from expected:    110
side 2 had  142204  occurrences       difference from expected:   -653
side 3 had  142993  occurrences       difference from expected:    136
side 4 had  142797  occurrences       difference from expected:    -60
side 5 had  143081  occurrences       difference from expected:    224
side 6 had  142711  occurrences       difference from expected:   -146
side 7 had  143247  occurrences       difference from expected:    390
```

## Ring

```# Project : Seven-sided dice from five-sided dice

for n = 1 to 20
d = dice7()
see "" + d + " "
next
see nl

func dice7()
x = dice5() * 5 + dice5() - 6
if x > 20
return dice7()
ok
dc = x % 7 + 1
return dc

func dice5()
rnd = random(4) + 1
return rnd```

Output:

```7 6 3 5 2 2 7 1 2 7 3 7 4 4 4 2 3 2 6 1
```

## RPL

`UNIF?` is defined at Verify distribution uniformity/Naive

Works with: Halcyon Calc version 4.2.7
```≪ ≪ RAND 5 * CEIL ≫ → dice5
≪ WHILE
dice5 EVAL 5 *
dice5 EVAL 6 - +
DUP 21 ≥
REPEAT DROP END
7 MOD 1 +
≫ ≫ 'DICE7' STO
```
```≪ DICE7 ≫ 100000 .1 UNIF?
```
Output:
```1: [ 14557 14245 14255 14400 14224 14151 14168 ]
```

Watchdog timer limits the loop to 100,000 items.

## Ruby

Translation of: Tcl

Uses `distcheck` from here.

```require './distcheck.rb'

def d5
1 + rand(5)
end

def d7
loop do
d55 = 5*d5 + d5 - 6
return (d55 % 7 + 1) if d55 < 21
end
end

distcheck(1_000_000) {d5}
distcheck(1_000_000) {d7}
```
Output:
```1 200227
2 200264
3 199777
4 199387
5 200345
1 143175
2 143031
3 142731
4 142716
5 142931
6 142605
7 142811```

## Scala

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

```import scala.util.Random

object SevenSidedDice extends App {
private val rnd = new Random

private def seven = {
var v = 21

def five = 1 + rnd.nextInt(5)

while (v > 20) v = five + five * 5 - 6
1 + v % 7
}

println("Random number from 1 to 7: " + seven)

}
```

## Sidef

Translation of: Perl
```func dice5 { 1 + 5.rand.int }

func dice7 {
loop {
var d7 = ((5*dice5() + dice5() - 6) % 8);
d7 && return d7;
}
}

var count7 = Hash.new;

var n = 1e6;
n.times { count7{dice7()} := 0 ++ }
count7.keys.sort.each { |k|
printf("%s: %5.2f%%\n", k, 100*(count7{k}/n * 7 - 1));
}
```
Output:
```1: -0.00%
2:  0.02%
3:  0.23%
4:  0.42%
5: -0.23%
6: -0.54%
7:  0.10%```

## Tcl

Any old D&D hand will know these as a D5 and a D7...

```proc D5 {} {expr {1 + int(5 * rand())}}

proc D7 {} {
while 1 {
set d55 [expr {5 * [D5] + [D5] - 6}]
if {\$d55 < 21} {
return [expr {\$d55 % 7 + 1}]
}
}
}
```

Checking:

```% distcheck D5 1000000
1 199893 2 200162 3 200075 4 199630 5 200240
% distcheck D7 1000000
1 143121 2 142383 3 143353 4 142811 5 142172 6 143291 7 142869
```

## VBA

The original StackOverflow page doesn't exist any longer. Luckily archive.org exists.

```Private Function Test4DiscreteUniformDistribution(ObservationFrequencies() As Variant, Significance As Single) As Boolean
'Returns true if the observed frequencies pass the Pearson Chi-squared test at the required significance level.
Dim Total As Long, Ei As Long, i As Integer
Dim ChiSquared As Double, DegreesOfFreedom As Integer, p_value As Double
Debug.Print "[1] ""Data set:"" ";
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
Total = Total + ObservationFrequencies(i)
Debug.Print ObservationFrequencies(i); " ";
Next i
DegreesOfFreedom = UBound(ObservationFrequencies) - LBound(ObservationFrequencies)
'This is exactly the number of different categories minus 1
Ei = Total / (DegreesOfFreedom + 1)
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
ChiSquared = ChiSquared + (ObservationFrequencies(i) - Ei) ^ 2 / Ei
Next i
p_value = 1 - WorksheetFunction.ChiSq_Dist(ChiSquared, DegreesOfFreedom, True)
Debug.Print
Debug.Print "Chi-squared test for given frequencies"
Debug.Print "X-squared ="; Format(ChiSquared, "0.0000"); ", ";
Debug.Print "df ="; DegreesOfFreedom; ", ";
Debug.Print "p-value = "; Format(p_value, "0.0000")
Test4DiscreteUniformDistribution = p_value > Significance
End Function
Private Function Dice5() As Integer
Dice5 = Int(5 * Rnd + 1)
End Function
Private Function Dice7() As Integer
Dim i As Integer
Do
i = 5 * (Dice5 - 1) + Dice5
Loop While i > 21
Dice7 = i Mod 7 + 1
End Function
Sub TestDice7()
Dim i As Long, roll As Integer
Dim Bins(1 To 7) As Variant
For i = 1 To 1000000
roll = Dice7
Bins(roll) = Bins(roll) + 1
Next i
Debug.Print "[1] ""Uniform? "; Test4DiscreteUniformDistribution(Bins, 0.05); """"
End Sub
```
Output:
```[1] "Data set:"  142418   142898   142940   142573   143030   143139   143002
Chi-squared test for given frequencies
X-squared =2.8870, df = 6 , p-value = 0.8229
[1] "Uniform? True"

```

## VBScript

```Option Explicit

function dice5
dice5 = int(rnd*5) + 1
end function

function dice7
dim j
do
j = 5 * dice5 + dice5 - 6
loop until j < 21
dice7 = j mod 7 + 1
end function
```

## Verilog

```///////////////////////////////////////////////////////////////////////////////
/// seven_sided_dice_tb : (testbench)                                       ///
///      Check the distribution of the output of a seven sided dice circuit ///
///////////////////////////////////////////////////////////////////////////////
module seven_sided_dice_tb;
reg [31:0] freq[0:6];
reg        clk;
wire [2:0] dice_face;
reg        req;
wire       valid_roll;
integer    i;
initial begin
clk <= 0;
forever begin
#1;
clk <= ~clk;
end
end
initial begin
req <= 1'b1;
for(i = 0; i < 7; i = i + 1) begin
freq[i] <= 32'b0;
end
repeat(10) @(posedge clk);
repeat(7000000) begin
@(posedge clk);
while(~valid_roll) begin
@(posedge clk);
end
freq[dice_face] <= freq[dice_face] + 32'b1;
end
\$display("********************************************");
\$display("*** Seven sided dice distribution:          ");
\$display("    Theoretical distribution is an uniform  ");
\$display("    distribution with (1/7)-probability     ");
\$display("    for each possible outcome,              ");
\$display("  The experimental distribution is:          ");
for(i = 0; i < 7; i = i + 1) begin
if(freq[i] < 32'd1_000_000) begin
\$display("%d with probability 1/7 - (%d ppm)", i, (32'd1_000_000 - freq[i])/7);
end
else begin
\$display("%d with probability 1/7 + (%d ppm)", i, (freq[i] - 32'd1_000_000)/7);
end
end
\$finish;
end

seven_sided_dice DUT(
.clk(clk),
.req(req),
.valid_roll(valid_roll),
.dice_face(dice_face)
);
endmodule
///////////////////////////////////////////////////////////////////////////////
/// seven_sided_dice :                                                      ///
///      Synthsizeable module that using a 5 sided dice as a black box      ///
///      is able to reproduce the outcomes if a 7-sided dice                ///
///////////////////////////////////////////////////////////////////////////////
module seven_sided_dice(
input wire       clk,
input wire       req,
output reg       valid_roll,
output reg [2:0] dice_face
);
wire [2:0] face1;
wire [2:0] face2;
reg [4:0] combination;
reg req_p1;
reg req_p2;
reg req_p3;
always @(posedge clk) begin
req_p1 <= req;
req_p2 <= req_p1;
end
always @(posedge clk) begin
if(req_p1) begin
combination <= face1 + face2 + {face2, 2'b00};
end
if(req_p2) begin
case(combination)
5'd0,  5'd1,  5'd2: {valid_roll, dice_face} <= {1'b1, 3'd0};
5'd3,  5'd4,  5'd5: {valid_roll, dice_face} <= {1'b1, 3'd1};
5'd6,  5'd7,  5'd8: {valid_roll, dice_face} <= {1'b1, 3'd2};
5'd9, 5'd10, 5'd11: {valid_roll, dice_face} <= {1'b1, 3'd3};
5'd12, 5'd13, 5'd14: {valid_roll, dice_face} <= {1'b1, 3'd4};
5'd15, 5'd16, 5'd17: {valid_roll, dice_face} <= {1'b1, 3'd5};
5'd18, 5'd19, 5'd20: {valid_roll, dice_face} <= {1'b1, 3'd6};
default: valid_roll <= 1'b0;
endcase
end
end

five_sided_dice dice1(
.clk(clk),
.req(req),
.dice_face(face1)
);

five_sided_dice dice2(
.clk(clk),
.req(req),
.dice_face(face2)
);
endmodule

///////////////////////////////////////////////////////////////////////////////
/// five_sided_dice :                                                       ///
///      A model of the five sided dice component                           ///
///////////////////////////////////////////////////////////////////////////////
module five_sided_dice(
input wire clk,
input wire req,
output reg [2:0] dice_face
);
always @(posedge clk) begin
if(req) begin
dice_face  <= \$urandom % 5;
end
end
endmodule
```

Compiling with Icarus Verilog

```> iverilog seven-sided-dice.v -o seven-sided-dice
```

Running the test

```> vvp seven-sided-dice
********************************************
*** Seven sided dice distribution:
Theoretical distribution is an uniform
distribution with (1/7)-probability
for each possible outcome,
The experimental distribution is:
0 with probability 1/7 + (        67 ppm)
1 with probability 1/7 - (        47 ppm)
2 with probability 1/7 + (        92 ppm)
3 with probability 1/7 - (        17 ppm)
4 with probability 1/7 - (        36 ppm)
5 with probability 1/7 + (        51 ppm)
6 with probability 1/7 - (       109 ppm)
```

## Wren

Translation of: Kotlin
Library: Wren-sort
Library: Wren-fmt
```import "random" for Random
import "./sort" for Sort
import "./fmt" for Fmt

var r = Random.new()

var dice5 = Fn.new { r.int(1, 6) }

var dice7 = Fn.new {
while (true) {
var t = (dice5.call() - 1) * 5 + dice5.call() - 1
if (t < 21) return 1 + (t/3).floor
}
}

var checkDist = Fn.new { |gen, nRepeats, tolerance|
var occurs = {}
for (i in 1..nRepeats) {
var d = gen.call()
occurs[d] = occurs.containsKey(d) ? occurs[d] + 1 : 1
}
var expected = (nRepeats/occurs.count).floor
var maxError = (expected * tolerance / 100).floor
System.print("Repetitions = %(nRepeats), Expected = %(expected)")
System.print("Tolerance = %(tolerance)\%, Max Error = %(maxError)\n")
System.print("Integer   Occurrences   Error  Acceptable")
var f = "  \$d        \$5d      \$5d     \$s"
var allAcceptable = true
var cmp = Fn.new { |me1, me2| (me1.key - me2.key).sign }
occurs = occurs.toList
Sort.insertion(occurs, cmp)
for (me in occurs) {
var k = me.key
var v = me.value
var error = (v - expected).abs
var acceptable = (error <= maxError) ? "Yes" : "No"
if (acceptable == "No") allAcceptable = false
Fmt.print(f, k, v, error, acceptable)
}
System.print("\nAcceptable overall: %(allAcceptable ? "Yes" : "No")")
}

checkDist.call(dice7, 1400000, 0.5)
```
Output:
```Repetitions = 1400000, Expected = 200000
Tolerance = 0.5%, Max Error = 1000

Integer   Occurrences   Error  Acceptable
1        199744        256     Yes
2        199678        322     Yes
3        200254        254     Yes
4        199903         97     Yes
5        200080         80     Yes
6        200070         70     Yes
7        200271        271     Yes

Acceptable overall: Yes
```

## zkl

```var die5=(1).random.fp(6); // [1..5]
fcn die7{ while((r:=5*die5() + die5())>=27){} r/3-1 }

fcn rtest(N){ //test spread over [0..9]
dist:=L(0,0,0,0,0,0,0,0,0,0);
do(N){ dist[die7()]+=1 }
sum:=dist.sum();
dist=dist.apply('wrap(n){ "%.2f%%".fmt(n.toFloat()/sum*100) }).println();
}

println("Looking for ",100.0/7,"%");
rtest(0d1_000_000);```
Output:
```Looking for 14.2857%
L("0.00%","14.28%","14.36%","14.22%","14.26%","14.34%","14.33%","14.21%","0.00%","0.00%")
```