Nimber arithmetic
Nimber arithmetic 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.
The nimbers, also known as Grundy numbers, are the values of the heaps in the game of Nim. They have addition and multiplication operations, unrelated to the addition and multiplication of the integers. Both operations are defined recursively:
The nim-sum of two integers m and n, denoted m⊕n is given by
m⊕n=mex(m'⊕n, m⊕ n' : m'<m, n'<n),
where the mex function returns the smallest integer not in the set. More simply: collect all the nim-sums of m and numbers smaller than n, and all nim-sums of n with all numbers less than m and find the smallest number not in that set. Fortunately, this also turns out to be equal to the bitwise xor of the two.
The nim-product is also defined recursively:
m⊗n=mex([m'⊗n]⊕[m⊗n']⊕[m'⊗n'] : m'<m, n'<n)
The product is more complicated and time-consuming to evaluate, but there are a few facts which may help:
- The operators ⊕ and ⊗ are commutative and distributive
- the nim-product of a Fermat power (22k) and a smaller number is their ordinary product
- the nim-square of a Fermat power x is the ordinary product 3x/2
- Tasks
- Create nimber addition and multiplication tables up to at least 15
- Find the nim-sum and nim-product of two five digit integers of your choice
11l
F hpo2(n)
R n [&] (-n)
F lhpo2(n)
V q = 0
V m = hpo2(n)
L m % 2 == 0
m = m >> 1
q++
R q
F nimsum(Int x, Int y)
R x (+) y
F nimprod(Int x, Int y)
I x < 2 | y < 2
R x * y
V h = hpo2(x)
I x > h
R nimprod(h, y) (+) nimprod(x (+) h, y)
I hpo2(y) < y
R nimprod(y, x)
V (xp, yp) = (lhpo2(x), lhpo2(y))
V comp = xp [&] yp
I comp == 0
R x * y
h = hpo2(comp)
R nimprod(nimprod(x >> h, y >> h), 3 << (h - 1))
L(f, op) ((nimsum, ‘+’), (nimprod, ‘*’))
print(‘ ’op‘ |’, end' ‘’)
L(i) 16
print(‘#3’.format(i), end' ‘’)
print("\n--- "(‘-’ * 48))
L(i) 16
print(‘#2 |’.format(i), end' ‘’)
L(j) 16
print(‘#3’.format(f(i, j)), end' ‘’)
print()
print()
V (a, b) = (21508, 42689)
print(a‘ + ’b‘ = ’nimsum(a, b))
print(a‘ * ’b‘ = ’nimprod(a, b))- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
C
#include <stdio.h>
#include <stdint.h>
// highest power of 2 that divides a given number
uint32_t hpo2(uint32_t n) {
return n & -n;
}
// base 2 logarithm of the highest power of 2 dividing a given number
uint32_t lhpo2(uint32_t n) {
uint32_t q = 0, m = hpo2(n);
for (; m % 2 == 0; m >>= 1, ++q) {}
return q;
}
// nim-sum of two numbers
uint32_t nimsum(uint32_t x, uint32_t y) {
return x ^ y;
}
// nim-product of two numbers
uint32_t nimprod(uint32_t x, uint32_t y) {
if (x < 2 || y < 2)
return x * y;
uint32_t h = hpo2(x);
if (x > h)
return nimprod(h, y) ^ nimprod(x ^ h, y);
if (hpo2(y) < y)
return nimprod(y, x);
uint32_t xp = lhpo2(x), yp = lhpo2(y);
uint32_t comp = xp & yp;
if (comp == 0)
return x * y;
h = hpo2(comp);
return nimprod(nimprod(x >> h, y >> h), 3 << (h - 1));
}
void print_table(uint32_t n, char op, uint32_t(*func)(uint32_t, uint32_t)) {
printf(" %c |", op);
for (uint32_t a = 0; a <= n; ++a)
printf("%3d", a);
printf("\n--- -");
for (uint32_t a = 0; a <= n; ++a)
printf("---");
printf("\n");
for (uint32_t b = 0; b <= n; ++b) {
printf("%2d |", b);
for (uint32_t a = 0; a <= n; ++a)
printf("%3d", func(a, b));
printf("\n");
}
}
int main() {
print_table(15, '+', nimsum);
printf("\n");
print_table(15, '*', nimprod);
const uint32_t a = 21508, b = 42689;
printf("\n%d + %d = %d\n", a, b, nimsum(a, b));
printf("%d * %d = %d\n", a, b, nimprod(a, b));
return 0;
}
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
C++
#include <cstdint>
#include <functional>
#include <iomanip>
#include <iostream>
// highest power of 2 that divides a given number
uint32_t hpo2(uint32_t n) {
return n & -n;
}
// base 2 logarithm of the highest power of 2 dividing a given number
uint32_t lhpo2(uint32_t n) {
uint32_t q = 0, m = hpo2(n);
for (; m % 2 == 0; m >>= 1, ++q) {}
return q;
}
// nim-sum of two numbers
uint32_t nimsum(uint32_t x, uint32_t y) {
return x ^ y;
}
// nim-product of two numbers
uint32_t nimprod(uint32_t x, uint32_t y) {
if (x < 2 || y < 2)
return x * y;
uint32_t h = hpo2(x);
if (x > h)
return nimprod(h, y) ^ nimprod(x ^ h, y);
if (hpo2(y) < y)
return nimprod(y, x);
uint32_t xp = lhpo2(x), yp = lhpo2(y);
uint32_t comp = xp & yp;
if (comp == 0)
return x * y;
h = hpo2(comp);
return nimprod(nimprod(x >> h, y >> h), 3 << (h - 1));
}
void print_table(uint32_t n, char op, std::function<uint32_t(uint32_t, uint32_t)> func) {
std::cout << ' ' << op << " |";
for (uint32_t a = 0; a <= n; ++a)
std::cout << std::setw(3) << a;
std::cout << "\n--- -";
for (uint32_t a = 0; a <= n; ++a)
std::cout << "---";
std::cout << '\n';
for (uint32_t b = 0; b <= n; ++b) {
std::cout << std::setw(2) << b << " |";
for (uint32_t a = 0; a <= n; ++a)
std::cout << std::setw(3) << func(a, b);
std::cout << '\n';
}
}
int main() {
print_table(15, '+', nimsum);
printf("\n");
print_table(15, '*', nimprod);
const uint32_t a = 21508, b = 42689;
std::cout << '\n';
std::cout << a << " + " << b << " = " << nimsum(a, b) << '\n';
std::cout << a << " * " << b << " = " << nimprod(a, b) << '\n';
return 0;
}
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Delphi
program Nimber_arithmetic;
uses
System.SysUtils, System.Math;
Type
TFnop = record
fn: TFunc<Cardinal, Cardinal, Cardinal>;
op: string;
end;
// Highest power of two that divides a given number.
function hpo2(n: Cardinal): Cardinal;
begin
Result := n and (-n)
end;
// Base 2 logarithm of the highest power of 2 dividing a given number.
function lhpo2(n: Cardinal): Cardinal;
var
m: Cardinal;
begin
Result := 0;
m := hpo2(n);
while m mod 2 = 0 do
begin
m := m shr 1;
inc(Result);
end;
end;
// nim-sum of two numbers.
function nimsum(x, y: Cardinal): Cardinal;
begin
Result := x xor y;
end;
function nimprod(x, y: Cardinal): Cardinal;
var
h, xp, yp, comp: Cardinal;
begin
if (x < 2) or (y < 2) then
exit(x * y);
h := hpo2(x);
if x > h then
exit((nimprod(h, y) xor nimprod((x xor h), y)));
if hpo2(y) < y then
exit(nimprod(y, x)); // break y into powers of 2 by flipping operands
xp := lhpo2(x);
yp := lhpo2(y);
comp := xp and yp;
if comp = 0 then
exit(x * y); // no Fermat power in common
h := hpo2(comp);
// a Fermat number square is its sequimultiple
Result := nimprod(nimprod(x shr h, y shr h), 3 shl (h - 1));
end;
var
fnop: array [0 .. 1] of TFnop;
f: TFnop;
i, j, a, b: Cardinal;
begin
with fnop[0] do
begin
fn := nimsum;
op := '+';
end;
with fnop[1] do
begin
fn := nimprod;
op := '*';
end;
for f in fnop do
begin
write(' ', f.op, ' |');
for i := 0 to 15 do
Write(i:3);
Writeln;
Writeln('--- ', string.Create('-', 48));
for i := 0 to 15 do
begin
write(i:2, ' |');
for j := 0 to 15 do
write(f.fn(i, j):3);
Writeln;
end;
Writeln;
end;
a := 21508;
b := 42689;
Writeln(Format('%d + %d = %d', [a, b, nimsum(a, b)]));
Writeln(Format('%d * %d = %d', [a, b, nimprod(a, b)]));
readln;
end.
Factor
USING: combinators formatting io kernel locals math sequences ;
! highest power of 2 that divides a given number
: hpo2 ( n -- n ) dup neg bitand ;
! base 2 logarithm of the highest power of 2 dividing a given number
: lhpo2 ( n -- n )
hpo2 0 swap [ dup even? ] [ -1 shift [ 1 + ] dip ] while drop ;
! nim sum of two numbers
ALIAS: nim-sum bitxor
! nim product of two numbers
:: nim-prod ( x y -- prod )
x hpo2 :> h!
0 :> comp!
{
{ [ x 2 < y 2 < or ] [ x y * ] }
{ [ x h > ] [ h y nim-prod x h bitxor y nim-prod bitxor ] } ! recursively break x into its powers of 2
{ [ y hpo2 y < ] [ y x nim-prod ] } ! recursively break y into its powers of 2 by flipping the operands
{ [ x y [ lhpo2 ] bi@ bitand comp! comp zero? ] [ x y * ] } ! we have no fermat power in common
[
comp hpo2 h! ! a fermat number square is its sequimultiple
x h neg shift y h neg shift nim-prod
3 h 1 - shift nim-prod
]
} cond ;
! words for printing tables
: dashes ( n -- ) [ CHAR: - ] "" replicate-as write ;
: top1 ( str -- ) " %s |" printf 16 <iota> [ "%3d" printf ] each nl ;
: top2 ( -- ) 3 dashes bl 49 dashes nl ;
: row ( n quot -- )
over "%2d |" printf curry 16 <iota> swap
[ call "%3d" printf ] curry each ; inline
: table ( quot str -- )
top1 top2 16 <iota> swap [ row nl ] curry each ; inline
! task
[ nim-sum ] "+" table nl
[ nim-prod ] "*" table nl
33333 77777
[ 2dup nim-sum "%d + %d = %d\n" printf ]
[ 2dup nim-prod "%d * %d = %d\n" printf ] 2bi
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 33333 + 77777 = 110052 33333 * 77777 = 2184564070
FreeBASIC
function hpo2( n as uinteger ) as uinteger
'highest power of 2 that divides a given number
return n and -n
end function
function lhpo2( n as uinteger ) as uinteger
'base 2 logarithm of the highest power of 2 dividing a given number
dim as uinteger q = 0, m = hpo2( n )
while m mod 2 = 0
m = m shr 1
q += 1
wend
return q
end function
function nimsum(x as uinteger, y as uinteger) as uinteger
'nim-sum of two numbers
return x xor y
end function
function nimprod(x as uinteger, y as uinteger) as uinteger
'nim-product of two numbers
if x < 2 orelse y < 2 then return x*y
dim as uinteger h = hpo2(x)
if x > h then return nimprod(h, y) xor nimprod(x xor h, y) 'recursively break x into its powers of 2
if hpo2(y) < y then return nimprod(y, x) 'recursively break y into its powers of 2 by flipping the operands
'now both x and y are powers of two
dim as uinteger xp = lhpo2(x), yp = lhpo2(y), comp = xp and yp
if comp = 0 then return x*y 'we have no fermat power in common
h = hpo2(comp)
return nimprod(nimprod(x shr h, y shr h), 3 shl (h - 1)) 'a fermat number square is its sequimultiple
end function
'print tables
function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
dim as uinteger a, b
dim as string outstr
outstr = " + | "
for a = 0 to 15
outstr += padto(2, a)+" "
next a
print outstr
print "--- -------------------------------------------------"
for b = 0 to 15
outstr = padto(2, b)+ " | "
for a = 0 to 15
outstr += padto(2, nimsum(a,b))+" "
next a
print outstr
next b
print
outstr = " * | "
for a = 0 to 15
outstr += padto(2, a)+" "
next a
print outstr
print "--- -------------------------------------------------"
for b = 0 to 15
outstr = padto(2, b)+ " | "
for a = 0 to 15
outstr += padto(2, nimprod(a,b))+" "
next a
print outstr
next b
print
a = 21508
b = 42689
print using "##### + ##### = ##########"; a; b; nimsum(a,b)
print using "##### * ##### = ##########"; a; b; nimprod(a,b)
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Go
package main
import (
"fmt"
"strings"
)
// Highest power of two that divides a given number.
func hpo2(n uint) uint { return n & (-n) }
// Base 2 logarithm of the highest power of 2 dividing a given number.
func lhpo2(n uint) uint {
q := uint(0)
m := hpo2(n)
for m%2 == 0 {
m = m >> 1
q++
}
return q
}
// nim-sum of two numbers.
func nimsum(x, y uint) uint { return x ^ y }
// nim-product of two numbers.
func nimprod(x, y uint) uint {
if x < 2 || y < 2 {
return x * y
}
h := hpo2(x)
if x > h {
return nimprod(h, y) ^ nimprod(x^h, y) // break x into powers of 2
}
if hpo2(y) < y {
return nimprod(y, x) // break y into powers of 2 by flipping operands
}
xp, yp := lhpo2(x), lhpo2(y)
comp := xp & yp
if comp == 0 {
return x * y // no Fermat power in common
}
h = hpo2(comp)
// a Fermat number square is its sequimultiple
return nimprod(nimprod(x>>h, y>>h), 3<<(h-1))
}
type fnop struct {
fn func(x, y uint) uint
op string
}
func main() {
for _, f := range []fnop{{nimsum, "+"}, {nimprod, "*"}} {
fmt.Printf(" %s |", f.op)
for i := 0; i <= 15; i++ {
fmt.Printf("%3d", i)
}
fmt.Println("\n--- " + strings.Repeat("-", 48))
for i := uint(0); i <= 15; i++ {
fmt.Printf("%2d |", i)
for j := uint(0); j <= 15; j++ {
fmt.Printf("%3d", f.fn(i, j))
}
fmt.Println()
}
fmt.Println()
}
a := uint(21508)
b := uint(42689)
fmt.Printf("%d + %d = %d\n", a, b, nimsum(a, b))
fmt.Printf("%d * %d = %d\n", a, b, nimprod(a, b))
}
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
J
nadd=: 22 b. NB. bitwise exclusive or on integers
and=: 17 b. NB. bitwise exclusive or on integers
nmul=: {{
if. x +.&(2&>) y do.
x*y
elseif. 1 < #_ q: x do.
h=. (and-) x
(h nmul y) nadd y nmul h nadd x
elseif. 1 < #_ q: y do.
y nmul x
else.
comp=. x and&(0 { 1 q: ]) y
if. 0=comp do.
x*y
else.
p=. 2^(and-) comp
(3*p%2) nmul x nmul&(%&p) y
end.
end.
}}M."0
Task examples:
nadd table _4+i.20
┌────┬───────────────────────────────────────────────────────────────────────┐
│nadd│ _4 _3 _2 _1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15│
├────┼───────────────────────────────────────────────────────────────────────┤
│_4 │ 0 1 2 3 _4 _3 _2 _1 _8 _7 _6 _5 _12 _11 _10 _9 _16 _15 _14 _13│
│_3 │ 1 0 3 2 _3 _4 _1 _2 _7 _8 _5 _6 _11 _12 _9 _10 _15 _16 _13 _14│
│_2 │ 2 3 0 1 _2 _1 _4 _3 _6 _5 _8 _7 _10 _9 _12 _11 _14 _13 _16 _15│
│_1 │ 3 2 1 0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _10 _11 _12 _13 _14 _15 _16│
│ 0 │ _4 _3 _2 _1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15│
│ 1 │ _3 _4 _1 _2 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14│
│ 2 │ _2 _1 _4 _3 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13│
│ 3 │ _1 _2 _3 _4 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12│
│ 4 │ _8 _7 _6 _5 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11│
│ 5 │ _7 _8 _5 _6 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10│
│ 6 │ _6 _5 _8 _7 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9│
│ 7 │ _5 _6 _7 _8 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8│
│ 8 │_12 _11 _10 _9 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7│
│ 9 │_11 _12 _9 _10 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6│
│10 │_10 _9 _12 _11 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5│
│11 │ _9 _10 _11 _12 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4│
│12 │_16 _15 _14 _13 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3│
│13 │_15 _16 _13 _14 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2│
│14 │_14 _13 _16 _15 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1│
│15 │_13 _14 _15 _16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0│
└────┴───────────────────────────────────────────────────────────────────────┘
nmul table _4+i.20
┌────┬───────────────────────────────────────────────────────────────────────────┐
│nmul│ _4 _3 _2 _1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15│
├────┼───────────────────────────────────────────────────────────────────────────┤
│_4 │ 16 12 8 4 0 _4 _8 _12 _16 _20 _24 _28 _32 _36 _40 _44 _48 _52 _56 _60│
│_3 │ 12 9 6 3 0 _3 _6 _9 _12 _15 _18 _21 _24 _27 _30 _33 _36 _39 _42 _45│
│_2 │ 8 6 4 2 0 _2 _4 _6 _8 _10 _12 _14 _16 _18 _20 _22 _24 _26 _28 _30│
│_1 │ 4 3 2 1 0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _10 _11 _12 _13 _14 _15│
│ 0 │ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0│
│ 1 │ _4 _3 _2 _1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15│
│ 2 │ _8 _6 _4 _2 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5│
│ 3 │_12 _9 _6 _3 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10│
│ 4 │_16 _12 _8 _4 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1│
│ 5 │_20 _15 _10 _5 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14│
│ 6 │_24 _18 _12 _6 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4│
│ 7 │_28 _21 _14 _7 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11│
│ 8 │_32 _24 _16 _8 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2│
│ 9 │_36 _27 _18 _9 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13│
│10 │_40 _30 _20 _10 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7│
│11 │_44 _33 _22 _11 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8│
│12 │_48 _36 _24 _12 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3│
│13 │_52 _39 _26 _13 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12│
│14 │_56 _42 _28 _14 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6│
│15 │_60 _45 _30 _15 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9│
└────┴───────────────────────────────────────────────────────────────────────────┘
12345 nadd 67890
80139
12345 nmul 67890
809054384
Java
import java.util.function.IntBinaryOperator;
public class Nimber {
public static void main(String[] args) {
printTable(15, '+', (x, y) -> nimSum(x, y));
System.out.println();
printTable(15, '*', (x, y) -> nimProduct(x, y));
System.out.println();
int a = 21508, b = 42689;
System.out.println(a + " + " + b + " = " + nimSum(a, b));
System.out.println(a + " * " + b + " = " + nimProduct(a, b));
}
// nim-sum of two numbers
public static int nimSum(int x, int y) {
return x ^ y;
}
// nim-product of two numbers
public static int nimProduct(int x, int y) {
if (x < 2 || y < 2)
return x * y;
int h = hpo2(x);
if (x > h)
return nimProduct(h, y) ^ nimProduct(x ^ h, y);
if (hpo2(y) < y)
return nimProduct(y, x);
int xp = lhpo2(x), yp = lhpo2(y);
int comp = xp & yp;
if (comp == 0)
return x * y;
h = hpo2(comp);
return nimProduct(nimProduct(x >> h, y >> h), 3 << (h - 1));
}
// highest power of 2 that divides a given number
private static int hpo2(int n) {
return n & -n;
}
// base 2 logarithm of the highest power of 2 dividing a given number
private static int lhpo2(int n) {
int q = 0, m = hpo2(n);
for (; m % 2 == 0; m >>= 1, ++q) {}
return q;
}
private static void printTable(int n, char op, IntBinaryOperator func) {
System.out.print(" " + op + " |");
for (int a = 0; a <= n; ++a)
System.out.printf("%3d", a);
System.out.print("\n--- -");
for (int a = 0; a <= n; ++a)
System.out.print("---");
System.out.println();
for (int b = 0; b <= n; ++b) {
System.out.printf("%2d |", b);
for (int a = 0; a <= n; ++a)
System.out.printf("%3d", func.applyAsInt(a, b));
System.out.println();
}
}
}
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Julia
""" highest power of 2 that divides a given number """
hpo2(n) = n & -n
""" base 2 logarithm of the highest power of 2 dividing a given number """
lhpo2(n) = begin q, m = 0, hpo2(n); while iseven(m) m >>= 1; q += 1 end; q end
""" nim-sum of two numbers """
nimsum(x, y) = x ⊻ y
""" nim-product of two numbers """
function nimprod(x, y)
(x < 2 || y < 2) && return x * y
h = hpo2(x)
(x > h) && return nimprod(h, y) ⊻ nimprod(x ⊻ h, y)
(hpo2(y) < y) && return nimprod(y, x)
xp, yp = lhpo2(x), lhpo2(y)
comp = xp & yp
comp == 0 && return x * y
h = hpo2(comp)
return nimprod(nimprod(x >> h, y >> h), 3 << (h - 1))
end
""" print a table of nim-sums or nim-products """
function printtable(n, op)
println(" $op |", prod([lpad(i, 3) for i in 0:n]), "\n--- -", "---"^(n + 1))
for j in 0:n
print(lpad(j, 2), " |")
for i in 0:n
print(lpad(op == '⊕' ? nimsum(i, j) : nimprod(i, j), 3))
end
print(j == n ? "\n\n" : "\n")
end
end
const a, b = 21508, 42689
printtable(15, '⊕')
printtable(15, '⊗')
println("nim-sum: $a ⊕ $b = $(nimsum(a, b))")
println("nim-product: $a ⊗ $b = $(nimprod(a, b))")
- Output:
⊕ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 ⊗ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 nim-sum: 21508 ⊕ 42689 = 62149 nim-product: 21508 ⊗ 42689 = 35202
Nim
import bitops, strutils
type Nimber = Natural
func hpo2(n: Nimber): Nimber =
## Return the highest power of 2 that divides a given number.
n and -n
func lhpo2(n: Nimber): Nimber =
## Return the base 2 logarithm of the highest power of 2 dividing a given number.
fastLog2(hpo2(n))
func ⊕(x, y: Nimber): Nimber =
## Return the nim-sum of two nimbers.
x xor y
func ⊗(x, y: Nimber): Nimber =
## Return the nim-product of two nimbers.
if x < 2 and y < 2: return x * y
var h = hpo2(x)
if x > h:
return ⊗(h, y) xor ⊗(x xor h, y) # Recursively break "x" into its powers of 2.
if hpo2(y) < y:
return ⊗(y, x) # Recursively break "y" into its powers of 2 by flipping the operands.
# Now both "x" and "y" are powers of two.
let comp = lhpo2(x) * lhpo2(y)
if comp == 0: return x * y # No Fermat number in common.
h = hpo2(comp)
# A fermat number square is its sequimultiple.
result = ⊗(⊗(x div (1 shl h), y div (1 shl h)), 3 * (1 shl (h - 1)))
when isMainModule:
for (opname, op) in [("⊕", ⊕), ("⊗", ⊗)]:
stdout.write ' ', opname, " |"
for i in 0..15: stdout.write ($i).align(3)
stdout.write "\n--- -", repeat('-', 48), '\n'
for b in 0..15:
stdout.write ($b).align(2), " |"
for a in 0..15:
stdout.write ($op(a, b)).align(3)
stdout.write '\n'
echo ""
const A = 21508
const B = 42689
echo "$1 ⊕ $2 = $3".format(A, B, ⊕(A, B))
echo "$1 ⊗ $2 = $3".format(A, B, ⊗(A, B))
- Output:
⊕ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 ⊗ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 ⊕ 42689 = 62149 21508 ⊗ 42689 = 35202
Perl
use strict;
use warnings;
use feature 'say';
use Math::AnyNum qw(:overload);
sub msb {
my($n, $base) = (shift, 0);
$base++ while $n >>= 1;
$base;
}
sub lsb {
my $n = shift;
msb($n & -$n);
}
sub nim_sum {
my($x,$y) = @_;
$x ^ $y
}
sub nim_prod {
no warnings qw(recursion);
my($x,$y) = @_;
return $x * $y if $x < 2 or $y < 2;
my $h = 2 ** lsb($x);
return nim_sum( nim_prod($h, $y), nim_prod(nim_sum($x,$h), $y)) if $x > $h;
return nim_prod($y,$x) if lsb($y) < msb($y);
return $x * $y unless my $comp = lsb($x) & lsb($y);
$h = 2 ** lsb($comp);
nim_prod(nim_prod(($x >> $h),($y >> $h)), (3 << ($h - 1)));
}
my $upto = 15;
for (['+', \&nim_sum], ['*', \&nim_prod]) {
my($op, $f) = @$_;
print " $op |"; printf '%3d', $_ for 0..$upto;
say "\n───┼" . ('────' x ($upto-3));
for my $r (0..$upto) {
printf('%2s |', $r);
printf '%3s', &$f($r, $_) for 0..$upto;
print "\n";
}
print "\n";
}
say nim_sum(21508, 42689);
say nim_prod(21508, 42689);
say nim_sum(2150821508215082150821508, 4268942689426894268942689);
say nim_prod(2150821508215082150821508, 4268942689426894268942689); # pretty slow
- Output:
+ │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ───┼──────────────────────────────────────────────── 0 │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 │ 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 │ 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 │ 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 │ 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 │ 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 │ 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 │ 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 │ 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 │ 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 │ 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 │ 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 │ 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 │ 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 │ 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 │ 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ───┼──────────────────────────────────────────────── 0 │ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 │ 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 │ 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 │ 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 │ 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 │ 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 │ 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 │ 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 │ 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 │ 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 │ 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 │ 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 │ 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 │ 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 │ 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 62149 35202 2722732241575131661744101 221974472829844568827862736061997038065
Phix
with javascript_semantics function hpo2(integer n) -- highest power of 2 that divides a given number return and_bits(n,-n) end function function lhpo2(integer n) -- base 2 logarithm of the highest power of 2 dividing a given number integer q = 0, m = hpo2(n) while remainder(m,2)=0 do m = floor(m/2) q += 1 end while return q end function function nimsum(integer x, y) -- nim-sum of two numbers return xor_bits(x,y) end function function nimprod(integer x, y) -- nim-product of two numbers if x < 2 or y < 2 then return x*y end if integer h = hpo2(x) if x > h then return xor_bits(nimprod(h, y),nimprod(xor_bits(x,h), y)) -- recursively break x into its powers of 2 elsif hpo2(y) < y then return nimprod(y, x) -- recursively break y into its powers of 2 by flipping the operands end if -- now both x and y are powers of two integer xp = lhpo2(x), yp = lhpo2(y), comp = and_bits(xp,yp) if comp = 0 then return x*y end if -- we have no fermat power in common h = hpo2(comp) return nimprod(nimprod(floor(x/power(2,h)), floor(y/power(2,h))), 3*power(2,h-1)) -- a fermat number square is its sequimultiple end function procedure print_table(integer n, op) -- print a table of nim-sums or nim-products printf(1," %c | "&join(repeat("%3d",n+1))&"\n",op&tagset(n,0)) printf(1,"---+%s\n",repeat('-',(n+1)*4)) for j=0 to n do printf(1,"%2d |",j) for i=0 to n do printf(1,"%4d",iff(op='+' ? nimsum(i, j) : nimprod(i, j))) end for printf(1,"\n") end for printf(1,"\n") end procedure print_table(25, '+') print_table(25, '*') constant a = 21508, b = 42689 printf(1,"%5d + %5d = %5d\n",{a,b,nimsum(a,b)}) printf(1,"%5d * %5d = %5d\n",{a,b,nimprod(a,b)})
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ---+-------------------------------------------------------------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20 23 22 25 24 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 23 20 21 26 27 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 19 18 17 16 23 22 21 20 27 26 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 17 18 19 28 29 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 21 20 23 22 17 16 19 18 29 28 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 19 16 17 30 31 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 23 22 21 20 19 18 17 16 31 30 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 29 30 31 16 17 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 25 24 27 26 29 28 31 30 17 16 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 31 28 29 18 19 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 27 26 25 24 31 30 29 28 19 18 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 25 26 27 20 21 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 29 28 31 30 25 24 27 26 21 20 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 27 24 25 22 23 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31 30 29 28 27 26 25 24 23 22 16 | 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 17 | 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 1 0 3 2 5 4 7 6 9 8 18 | 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 7 4 5 10 11 19 | 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 3 2 1 0 7 6 5 4 11 10 20 | 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 1 2 3 12 13 21 | 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 5 4 7 6 1 0 3 2 13 12 22 | 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 6 7 4 5 2 3 0 1 14 15 23 | 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 7 6 5 4 3 2 1 0 15 14 24 | 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 8 9 10 11 12 13 14 15 0 1 25 | 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 9 8 11 10 13 12 15 14 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ---+-------------------------------------------------------------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 32 34 35 33 40 42 43 41 44 46 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 48 51 49 50 60 63 61 62 52 55 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 64 68 72 76 70 66 78 74 75 79 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 80 85 90 95 82 87 88 93 83 86 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 96 102 107 109 110 104 101 99 103 97 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 112 119 121 126 122 125 115 116 127 120 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 128 136 140 132 139 131 135 143 141 133 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 144 153 158 151 159 150 145 152 149 156 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 160 170 175 165 163 169 172 166 161 171 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 176 187 189 182 183 188 186 177 185 178 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 192 204 196 200 205 193 201 197 198 202 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 208 221 214 219 217 212 223 210 222 211 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 224 238 231 233 229 235 226 236 234 228 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 240 255 245 250 241 254 244 251 242 253 16 | 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 24 8 56 40 88 72 120 104 152 136 17 | 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 8 25 42 59 76 93 110 127 128 145 18 | 0 18 35 49 72 90 107 121 140 158 175 189 196 214 231 245 56 42 27 9 112 98 83 65 180 166 19 | 0 19 33 50 76 95 109 126 132 151 165 182 200 219 233 250 40 59 9 26 100 119 69 86 172 191 20 | 0 20 40 60 70 82 110 122 139 159 163 183 205 217 229 241 88 76 112 100 30 10 54 34 211 199 21 | 0 21 42 63 66 87 104 125 131 150 169 188 193 212 235 254 72 93 98 119 10 31 32 53 203 222 22 | 0 22 43 61 78 88 101 115 135 145 172 186 201 223 226 244 120 110 83 69 54 32 29 11 255 233 23 | 0 23 41 62 74 93 99 116 143 152 166 177 197 210 236 251 104 127 65 86 34 53 11 28 231 240 24 | 0 24 44 52 75 83 103 127 141 149 161 185 198 222 234 242 152 128 180 172 211 203 255 231 21 13 25 | 0 25 46 55 79 86 97 120 133 156 171 178 202 211 228 253 136 145 166 191 199 222 233 240 13 20 21508 + 42689 = 62149 21508 * 42689 = 35202
Prolog
% highest power of 2 that divides a given number
hpo2(N, P):-
P is N /\ -N.
% base 2 logarithm of the highest power of 2 dividing a given number
lhpo2(N, Q):-
hpo2(N, M),
lhpo2_(M, 0, Q).
lhpo2_(M, Q, Q):-
1 is M mod 2,
!.
lhpo2_(M, Q1, Q):-
M1 is M >> 1,
Q2 is Q1 + 1,
lhpo2_(M1, Q2, Q).
% nim-sum of two numbers
nimsum(X, Y, Sum):-
Sum is X xor Y.
% nim-product of twp numbers
nimprod(X, Y, Product):-
(X < 2 ; Y < 2),
!,
Product is X * Y.
nimprod(X, Y, Product):-
hpo2(X, H),
X > H,
!,
nimprod(H, Y, P1),
X1 is X xor H,
nimprod(X1, Y, P2),
Product is P1 xor P2.
nimprod(X, Y, Product):-
hpo2(Y, H),
H < Y,
!,
nimprod(Y, X, Product).
nimprod(X, Y, Product):-
lhpo2(X, Xp),
lhpo2(Y, Yp),
Comp is Xp /\ Yp,
(Comp == 0 ->
Product is X * Y
;
hpo2(Comp, H),
X1 is X >> H,
Y1 is Y >> H,
Z is 3 << (H - 1),
nimprod(X1, Y1, P),
nimprod(P, Z, Product)
).
print_row(N, B, Function):-
writef('%3r |', [B]),
Goal =.. [Function, A, B, C],
forall(between(0, N, A), (Goal, writef('%3r', [C]))),
nl.
print_table(N, Operator, Function):-
writef(' %w |', [Operator]),
forall(between(0, N, A), writef('%3r', [A])),
writef('\n --- -', []),
forall(between(0, N, _), writef('---', [])),
nl,
forall(between(0, N, A), print_row(N, A, Function)).
main:-
print_table(15, '+', nimsum),
nl,
print_table(15, '*', nimprod),
nl,
A = 21508, B = 42689,
nimsum(A, B, Sum),
nimprod(A, B, Product),
writef('%w + %w = %w\n', [A, B, Sum]),
writef('%w * %w = %w\n', [A, B, Product]).
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Python
# Highest power of two that divides a given number.
def hpo2(n): return n & (-n)
# Base 2 logarithm of the highest power of 2 dividing a given number.
def lhpo2(n):
q = 0
m = hpo2(n)
while m%2 == 0:
m = m >> 1
q += 1
return q
def nimsum(x,y): return x ^ y
def nimprod(x,y):
if x < 2 or y < 2:
return x * y
h = hpo2(x)
if x > h:
return nimprod(h, y) ^ nimprod(x^h, y) # break x into powers of 2
if hpo2(y) < y:
return nimprod(y, x) # break y into powers of 2 by flipping operands
xp, yp = lhpo2(x), lhpo2(y)
comp = xp & yp
if comp == 0:
return x * y # no Fermat power in common
h = hpo2(comp)
# a Fermat number square is its sequimultiple
return nimprod(nimprod(x>>h, y>>h), 3<<(h-1))
if __name__ == '__main__':
for f, op in ((nimsum, '+'), (nimprod, '*')):
print(f" {op} |", end='')
for i in range(16):
print(f"{i:3d}", end='')
print("\n--- " + "-"*48)
for i in range(16):
print(f"{i:2d} |", end='')
for j in range(16):
print(f"{f(i,j):3d}", end='')
print()
print()
a, b = 21508, 42689
print(f"{a} + {b} = {nimsum(a,b)}")
print(f"{a} * {b} = {nimprod(a,b)}")
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Quackery
(Mostly translated from Julia, although 'translated' doesn't do the process justice.)
[ dup negate & ] is hpo2 ( n --> n )
[ 1 & 0 = ] is even ( n --> b )
[ 0 swap hpo2
[ dup even while
1 >>
dip 1+ again ]
drop ] is lhpo2 ( n --> n )
[ ^ ] is nim+ ( n n --> n )
forward is nim* ( x y --> r )
[ over 2 < over 2 < or iff
* done
over dup hpo2
tuck > iff
[ 2dup swap nim*
dip [ rot nim+ swap nim* ]
nim+ ] done
drop
dup hpo2 over < iff
[ swap nim* ] done
over lhpo2 over lhpo2 &
dup 0 = iff
[ drop * ] done
hpo2 tuck >>
dip [ tuck >> ]
nim*
swap 1 - 3 swap <<
nim* ] resolves nim* ( x y --> r )
[ over size -
space swap of
swap join ] is justify ( $ n --> $ )
[ number$
3 justify
echo$ ] is j.echo ( n --> )
[ cr sp echo$ say "|"
temp put
16 times [ i^ j.echo ] cr
sp char - 3 of echo$
say "+"
char - 48 of echo$ cr
16 times
[ i^ dup j.echo
say " |"
16 times
[ dup i^
temp share do
j.echo ]
drop cr ]
temp release ] is tabulate ( $ x --> )
' nim+ $ "(+)" tabulate
cr
' nim* $ "(*)" tabulate
cr
say " 10547 (+) 14447 = " 10547 14447 nim+ echo cr
say " 10547 (*) 14447 = " 10547 14447 nim* echo cr- Output:
(+)| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ---+------------------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 (*)| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ---+------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 10547 (+) 14447 = 4444 10547 (*) 14447 = 4444
Raku
(or at least, heavily inspired by FreeBasic)
Not limited by integer size. Doesn't rely on twos complement bitwise and.
sub infix:<⊕> (Int $x, Int $y) { $x +^ $y }
sub infix:<⊗> (Int $x, Int $y) {
return $x × $y if so $x|$y < 2;
my $h = exp $x.lsb, 2;
return ($h ⊗ $y) ⊕ (($x ⊕ $h) ⊗ $y) if $x > $h;
return ($y ⊗ $x) if $y.lsb < $y.msb;
return $x × $y unless my $comp = $x.lsb +& $y.lsb;
$h = exp $comp.lsb, 2;
(($x +> $h) ⊗ ($y +> $h)) ⊗ (3 +< ($h - 1))
}
# TESTING
my $upto = 26;
for <⊕>, &infix:<⊕>,
<⊗>, &infix:<⊗>
-> $op, &f {
put " $op │", ^$upto .fmt('%3s'), "\n───┼", '────' x $upto;
-> $r { put $r.fmt('%2s'), ' │', ^$upto .map: { &f($r, $_).fmt('%3s')} } for ^$upto;
put "\n";
}
put "21508 ⊕ 42689 = ", 21508 ⊕ 42689;
put "21508 ⊗ 42689 = ", 21508 ⊗ 42689;
put "2150821508215082150821508 ⊕ 4268942689426894268942689 = ", 2150821508215082150821508 ⊕ 4268942689426894268942689;
put "2150821508215082150821508 ⊗ 4268942689426894268942689 = ", 2150821508215082150821508 ⊗ 4268942689426894268942689;
- Output:
⊕ │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ───┼──────────────────────────────────────────────────────────────────────────────────────────────────────── 0 │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 │ 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20 23 22 25 24 2 │ 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 23 20 21 26 27 3 │ 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 19 18 17 16 23 22 21 20 27 26 4 │ 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 17 18 19 28 29 5 │ 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 21 20 23 22 17 16 19 18 29 28 6 │ 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 19 16 17 30 31 7 │ 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 23 22 21 20 19 18 17 16 31 30 8 │ 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 29 30 31 16 17 9 │ 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 25 24 27 26 29 28 31 30 17 16 10 │ 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 31 28 29 18 19 11 │ 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 27 26 25 24 31 30 29 28 19 18 12 │ 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 25 26 27 20 21 13 │ 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 29 28 31 30 25 24 27 26 21 20 14 │ 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 27 24 25 22 23 15 │ 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31 30 29 28 27 26 25 24 23 22 16 │ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 17 │ 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 1 0 3 2 5 4 7 6 9 8 18 │ 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 7 4 5 10 11 19 │ 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 3 2 1 0 7 6 5 4 11 10 20 │ 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 1 2 3 12 13 21 │ 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 5 4 7 6 1 0 3 2 13 12 22 │ 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 6 7 4 5 2 3 0 1 14 15 23 │ 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 7 6 5 4 3 2 1 0 15 14 24 │ 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 8 9 10 11 12 13 14 15 0 1 25 │ 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 9 8 11 10 13 12 15 14 1 0 ⊗ │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ───┼──────────────────────────────────────────────────────────────────────────────────────────────────────── 0 │ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 │ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 │ 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 32 34 35 33 40 42 43 41 44 46 3 │ 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 48 51 49 50 60 63 61 62 52 55 4 │ 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 64 68 72 76 70 66 78 74 75 79 5 │ 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 80 85 90 95 82 87 88 93 83 86 6 │ 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 96 102 107 109 110 104 101 99 103 97 7 │ 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 112 119 121 126 122 125 115 116 127 120 8 │ 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 128 136 140 132 139 131 135 143 141 133 9 │ 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 144 153 158 151 159 150 145 152 149 156 10 │ 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 160 170 175 165 163 169 172 166 161 171 11 │ 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 176 187 189 182 183 188 186 177 185 178 12 │ 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 192 204 196 200 205 193 201 197 198 202 13 │ 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 208 221 214 219 217 212 223 210 222 211 14 │ 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 224 238 231 233 229 235 226 236 234 228 15 │ 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 240 255 245 250 241 254 244 251 242 253 16 │ 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 24 8 56 40 88 72 120 104 152 136 17 │ 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 8 25 42 59 76 93 110 127 128 145 18 │ 0 18 35 49 72 90 107 121 140 158 175 189 196 214 231 245 56 42 27 9 112 98 83 65 180 166 19 │ 0 19 33 50 76 95 109 126 132 151 165 182 200 219 233 250 40 59 9 26 100 119 69 86 172 191 20 │ 0 20 40 60 70 82 110 122 139 159 163 183 205 217 229 241 88 76 112 100 30 10 54 34 211 199 21 │ 0 21 42 63 66 87 104 125 131 150 169 188 193 212 235 254 72 93 98 119 10 31 32 53 203 222 22 │ 0 22 43 61 78 88 101 115 135 145 172 186 201 223 226 244 120 110 83 69 54 32 29 11 255 233 23 │ 0 23 41 62 74 93 99 116 143 152 166 177 197 210 236 251 104 127 65 86 34 53 11 28 231 240 24 │ 0 24 44 52 75 83 103 127 141 149 161 185 198 222 234 242 152 128 180 172 211 203 255 231 21 13 25 │ 0 25 46 55 79 86 97 120 133 156 171 178 202 211 228 253 136 145 166 191 199 222 233 240 13 20 21508 ⊕ 42689 = 62149 21508 ⊗ 42689 = 35202 2150821508215082150821508 ⊕ 4268942689426894268942689 = 2722732241575131661744101 2150821508215082150821508 ⊗ 4268942689426894268942689 = 221974472829844568827862736061997038065
REXX
This REXX version optimizes the nimber product by using the nimber sum for some of its calculations.
The table size (for nimber sum and nimber products) may be specified on the command line (CL) as well as the
two test numbers.
/*REXX program performs nimber arithmetic (addition and multiplication); shows a table.*/
numeric digits 40; d= digits() % 8 /*use a big enough number of decimals. */
parse arg sz aa bb . /*obtain optional argument from the CL.*/
if sz=='' | sz=="," then sz= 15 /*Not specified? Then use the default.*/
if aa=='' | aa=="," then aa= 21508 /* " " " " " " */
if bb=='' | bb=="," then bb= 42689 /* " " " " " " */
w= max(4,length(sz)); @.= '+'; @.1= "*"; _= '═' /*calculate the width of the table cols*/
!= '║'; sz1= sz + 1; w1= w-1 /*define the "dash" character for table*/
do am=0 for 2 /*perform sums, then perform multiplies*/
call top ! || center("("@.am')', w1) /*show title of table. */
do j=0 for sz1; $= !||center(j, w1)! /*calculate & format a row of the table*/
do k=0 for sz1 /*build a row of table. */
if am then $= $ || right( nprod(j, k), w) /*append to a table row.*/
else $= $ || right( nsum(j, k), w) /* " " " " " */
end /*k*/
say $ ! /*show a row of a table.*/
end /*j*/
call bot
end /*am*/
say 'nimber sum of ' comma(aa) " and " comma(bb) ' ───► ' comma( nsum(aa, bb))
say 'nimber product of ' comma(aa) " and " comma(bb) ' ───► ' comma(nprod(aa, bb))
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
hdr: $= ? || !; do i=0 to sz; $=$ || right(i,w); end; say $ !; call sep; return
top: $= '╔'copies(_, w1)"╦"copies(copies(_, w), sz1)_; say $'╗'; arg ?; call hdr; return
sep: $= '╠'copies(_, w1)"╬"copies(copies(_, w), sz1)_; say $'╣'; return
bot: $= '╚'copies(_, w1)"╩"copies(copies(_, w), sz1)_; say $'╝'; say; say; return
comma: parse arg ?; do jc=length(?)-3 to 1 by -3; ?= insert(',', ?, jc); end; return ?
d2b: procedure; parse arg z; return right( x2b( d2x(z) ), digits(), 0)
hpo2: procedure; parse arg z; return 2 ** (length( d2b(z) + 0) - 1)
lhpo2: procedure; arg z; m=hpo2(z); q=0; do while m//2==0; m= m%2; q= q+1; end; return q
nsum: procedure expose d; parse arg x,y; return c2d( bitxor( d2c(x,d), d2c(y,d) ) )
shl: procedure; parse arg z,h; return z * (2**h)
shr: procedure; parse arg z,h; return z % (2**h)
/*──────────────────────────────────────────────────────────────────────────────────────*/
nprod: procedure expose d; parse arg x,y; if x<2 | y<2 then return x * y; h= hpo2(x)
if x>h then return nsum( nprod(h, y), nprod( nsum(x, h), y) )
if hpo2(y)<y then return nprod(y, x)
ands= c2d(bitand(d2c(lhpo2(x), d), d2c(lhpo2(y), d))); if ands==0 then return x*y
h= hpo2(ands); return nprod( nprod( shr(x,h), shr(y,h) ), shl(3, h-1) )
- output when using the input of: 25
╔═══╦═════════════════════════════════════════════════════════════════════════════════════════════════════════╗ ║(+)║ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ║ ╠═══╬═════════════════════════════════════════════════════════════════════════════════════════════════════════╣ ║ 0 ║ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ║ ║ 1 ║ 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20 23 22 25 24 ║ ║ 2 ║ 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 23 20 21 26 27 ║ ║ 3 ║ 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 19 18 17 16 23 22 21 20 27 26 ║ ║ 4 ║ 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 17 18 19 28 29 ║ ║ 5 ║ 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 21 20 23 22 17 16 19 18 29 28 ║ ║ 6 ║ 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 19 16 17 30 31 ║ ║ 7 ║ 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 23 22 21 20 19 18 17 16 31 30 ║ ║ 8 ║ 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 29 30 31 16 17 ║ ║ 9 ║ 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 25 24 27 26 29 28 31 30 17 16 ║ ║10 ║ 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 31 28 29 18 19 ║ ║11 ║ 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 27 26 25 24 31 30 29 28 19 18 ║ ║12 ║ 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 25 26 27 20 21 ║ ║13 ║ 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 29 28 31 30 25 24 27 26 21 20 ║ ║14 ║ 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 27 24 25 22 23 ║ ║15 ║ 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31 30 29 28 27 26 25 24 23 22 ║ ║16 ║ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 ║ ║17 ║ 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 1 0 3 2 5 4 7 6 9 8 ║ ║18 ║ 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 7 4 5 10 11 ║ ║19 ║ 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 3 2 1 0 7 6 5 4 11 10 ║ ║20 ║ 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 1 2 3 12 13 ║ ║21 ║ 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 5 4 7 6 1 0 3 2 13 12 ║ ║22 ║ 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 6 7 4 5 2 3 0 1 14 15 ║ ║23 ║ 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 7 6 5 4 3 2 1 0 15 14 ║ ║24 ║ 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 8 9 10 11 12 13 14 15 0 1 ║ ║25 ║ 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 9 8 11 10 13 12 15 14 1 0 ║ ╚═══╩═════════════════════════════════════════════════════════════════════════════════════════════════════════╝ ╔═══╦═════════════════════════════════════════════════════════════════════════════════════════════════════════╗ ║(*)║ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ║ ╠═══╬═════════════════════════════════════════════════════════════════════════════════════════════════════════╣ ║ 0 ║ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ║ ║ 1 ║ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ║ ║ 2 ║ 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 32 34 35 33 40 42 43 41 44 46 ║ ║ 3 ║ 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 48 51 49 50 60 63 61 62 52 55 ║ ║ 4 ║ 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 64 68 72 76 70 66 78 74 75 79 ║ ║ 5 ║ 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 80 85 90 95 82 87 88 93 83 86 ║ ║ 6 ║ 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 96 102 107 109 110 104 101 99 103 97 ║ ║ 7 ║ 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 112 119 121 126 122 125 115 116 127 120 ║ ║ 8 ║ 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 128 136 140 132 139 131 135 143 141 133 ║ ║ 9 ║ 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 144 153 158 151 159 150 145 152 149 156 ║ ║10 ║ 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 160 170 175 165 163 169 172 166 161 171 ║ ║11 ║ 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 176 187 189 182 183 188 186 177 185 178 ║ ║12 ║ 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 192 204 196 200 205 193 201 197 198 202 ║ ║13 ║ 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 208 221 214 219 217 212 223 210 222 211 ║ ║14 ║ 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 224 238 231 233 229 235 226 236 234 228 ║ ║15 ║ 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 240 255 245 250 241 254 244 251 242 253 ║ ║16 ║ 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 24 8 56 40 88 72 120 104 152 136 ║ ║17 ║ 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 8 25 42 59 76 93 110 127 128 145 ║ ║18 ║ 0 18 35 49 72 90 107 121 140 158 175 189 196 214 231 245 56 42 27 9 112 98 83 65 180 166 ║ ║19 ║ 0 19 33 50 76 95 109 126 132 151 165 182 200 219 233 250 40 59 9 26 100 119 69 86 172 191 ║ ║20 ║ 0 20 40 60 70 82 110 122 139 159 163 183 205 217 229 241 88 76 112 100 30 10 54 34 211 199 ║ ║21 ║ 0 21 42 63 66 87 104 125 131 150 169 188 193 212 235 254 72 93 98 119 10 31 32 53 203 222 ║ ║22 ║ 0 22 43 61 78 88 101 115 135 145 172 186 201 223 226 244 120 110 83 69 54 32 29 11 255 233 ║ ║23 ║ 0 23 41 62 74 93 99 116 143 152 166 177 197 210 236 251 104 127 65 86 34 53 11 28 231 240 ║ ║24 ║ 0 24 44 52 75 83 103 127 141 149 161 185 198 222 234 242 152 128 180 172 211 203 255 231 21 13 ║ ║25 ║ 0 25 46 55 79 86 97 120 133 156 171 178 202 211 228 253 136 145 166 191 199 222 233 240 13 20 ║ ╚═══╩═════════════════════════════════════════════════════════════════════════════════════════════════════════╝ nimber sum of 21,508 and 42,689 ───► 62,149 nimber product of 21,508 and 42,689 ───► 35,202
Rust
// highest power of 2 that divides a given number
fn hpo2(n: u32) -> u32 {
n & (0xFFFFFFFF - n + 1)
}
// base 2 logarithm of the highest power of 2 dividing a given number
fn lhpo2(n: u32) -> u32 {
let mut q: u32 = 0;
let mut m: u32 = hpo2(n);
while m % 2 == 0 {
m >>= 1;
q += 1;
}
q
}
// nim-sum of two numbers
fn nimsum(x: u32, y: u32) -> u32 {
x ^ y
}
// nim-product of two numbers
fn nimprod(x: u32, y: u32) -> u32 {
if x < 2 || y < 2 {
return x * y;
}
let mut h: u32 = hpo2(x);
if x > h {
return nimprod(h, y) ^ nimprod(x ^ h, y);
}
if hpo2(y) < y {
return nimprod(y, x);
}
let xp: u32 = lhpo2(x);
let yp: u32 = lhpo2(y);
let comp: u32 = xp & yp;
if comp == 0 {
return x * y;
}
h = hpo2(comp);
nimprod(nimprod(x >> h, y >> h), 3 << (h - 1))
}
fn print_table(n: u32, op: char, func: fn(u32, u32) -> u32) {
print!(" {} |", op);
for a in 0..=n {
print!("{:3}", a);
}
print!("\n--- -");
for _ in 0..=n {
print!("---");
}
println!();
for b in 0..=n {
print!("{:2} |", b);
for a in 0..=n {
print!("{:3}", func(a, b));
}
println!();
}
}
fn main() {
print_table(15, '+', nimsum);
println!();
print_table(15, '*', nimprod);
let a: u32 = 21508;
let b: u32 = 42689;
println!("\n{} + {} = {}", a, b, nimsum(a, b));
println!("{} * {} = {}", a, b, nimprod(a, b));
}
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Swift
import Foundation
// highest power of 2 that divides a given number
func hpo2(_ n: Int) -> Int {
n & -n
}
// base 2 logarithm of the highest power of 2 dividing a given number
func lhpo2(_ n: Int) -> Int {
var q: Int = 0
var m: Int = hpo2(n)
while m % 2 == 0 {
m >>= 1
q += 1
}
return q
}
// nim-sum of two numbers
func nimSum(x: Int, y: Int) -> Int {
x ^ y
}
// nim-product of two numbers
func nimProduct(x: Int, y: Int) -> Int {
if x < 2 || y < 2 {
return x * y
}
var h = hpo2(x);
if x > h {
return nimProduct(x: h, y: y) ^ nimProduct(x: x ^ h, y: y)
}
if hpo2(y) < y {
return nimProduct(x: y, y: x)
}
let xp = lhpo2(x)
let yp = lhpo2(y)
let comp = xp & yp
if comp == 0 {
return x * y
}
h = hpo2(comp)
return nimProduct(x: nimProduct(x: x >> h, y: y >> h), y: 3 << (h - 1))
}
func printTable(n: Int, op: Character, function: (Int, Int) -> Int) {
print(" \(op) |", terminator: "")
for a in 0...n {
print(String(format: "%3d", a), terminator: "")
}
print("\n--- -", terminator: "")
for _ in 0...n {
print("---", terminator: "")
}
print()
for b in 0...n {
print("\(String(format: "%2d", b)) |", terminator: "")
for a in 0...n {
print(String(format: "%3d", function(a, b)), terminator: "")
}
print()
}
}
printTable(n: 15, op: "+", function: nimSum)
print()
printTable(n: 15, op: "*", function: nimProduct)
let a: Int = 21508
let b: Int = 42689
print("\n\(a) + \(b) = \(nimSum(x: a, y: b))")
print("\(a) * \(b) = \(nimProduct(x: a, y: b))")
- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
Wren
import "./fmt" for Fmt
// Highest power of two that divides a given number.
var hpo2 = Fn.new { |n| n & (-n) }
// Base 2 logarithm of the highest power of 2 dividing a given number.
var lhpo2 = Fn.new { |n|
var q = 0
var m = hpo2.call(n)
while (m%2 == 0) {
m = m >> 1
q = q + 1
}
return q
}
// nim-sum of two numbers.
var nimsum = Fn.new { |x, y| x ^ y }
// nim-product of two numbers.
var nimprod // recursive
nimprod = Fn.new { |x, y|
if (x < 2 || y < 2) return x * y
var h = hpo2.call(x)
System.write("") // fixes VM recursion bug
if (x > h) return nimprod.call(h, y) ^ nimprod.call(x ^ h, y) // break x into powers of 2
if (hpo2.call(y) < y) return nimprod.call(y, x) // break y into powers of 2
var xp = lhpo2.call(x)
var yp = lhpo2.call(y)
var comp = xp & yp
if (comp == 0) return x * y // no Fermat power in common
h = hpo2.call(comp)
// a Fermat number square is its sequimultiple
return nimprod.call(nimprod.call(x >> h, y >> h), 3 << (h-1))
}
var fns = [[nimsum, "⊕"], [nimprod, "⊗"]]
for (fn in fns) {
System.write(" %(fn[1]) |")
for (i in 0..15) System.write(Fmt.d(3, i))
System.print("\n---+%("-" * 48)")
for (i in 0..15) {
System.write("%(Fmt.d(2, i)) |")
for (j in 0..15) System.write(Fmt.d(3, fn[0].call(i, j)))
System.print()
}
System.print()
}
var a = 21508
var b = 42689
System.print("%(a) + %(b) = %(nimsum.call(a, b))")
System.print("%(a) * %(b) = %(nimprod.call(a, b))")
- Output:
⊕ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ---+------------------------------------------------ 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 ⊗ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ---+------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 * 42689 = 35202
XPL0
include xpllib; \for Print
function HPo2(N); \Highest power of 2 that divides a given number
integer N;
return N and -N;
function LHPo2(N);
\Base 2 logarithm of the highest power of 2 dividing a given number
integer N, Q, M;
[Q:= 0; M:= HPo2(N);
while (M and 1) = 0 do
[M:= M >> 1;
Q:= Q+1;
];
return Q;
];
function NimSum(X, Y); \Nim-sum of two numbers
integer X, Y;
return X xor Y;
function NimProd(X, Y); \Nim-product of two numbers
integer X, Y, H, XP, YP, Comp;
[if X < 2 or Y < 2 then return X*Y;
H:= HPo2(X);
\Recursively break X into its powers of 2
if X > H then return NimProd(H, Y) xor NimProd(X xor H, Y);
\Recursively break Y into its powers of 2 by flipping its operands
if HPo2(Y) < Y then return NimProd(Y, X);
\Now both X and Y are powers of two
XP:= LHPo2(X); YP:= LHPo2(Y); Comp:= XP and YP;
if Comp = 0 then return X*Y; \there is no Fermat power in common
H:= HPo2(Comp);
\A Fermat number square is its sequimultiple
return NimProd(NimProd(X>>H, Y>>H), 3<<(H-1));
];
integer A, B;
[Format(3, 0);
Print(" + |");
for A:= 0 to 15 do RlOut(0, float(A));
Print("\n --- -------------------------------------------------\n");
for B:= 0 to 15 do
[RlOut(0, float(B));
Print(" |");
for A:= 0 to 15 do
RlOut(0, float(NimSum(A,B)));
Print("\n");
];
Print("\n * |");
for A:= 0 to 15 do RlOut(0, float(A));
Print("\n --- -------------------------------------------------\n");
for B:= 0 to 15 do
[RlOut(0, float(B));
Print(" |");
for A:= 0 to 15 do
RlOut(0, float(NimProd(A,B)));
Print("\n");
];
A:= 21508;
B:= 42689;
Print("\n%5d + %5d = %10d\n", A, B, NimSum(A,B));
Print("%5d + %5d = %10d\n", A, B, NimProd(A,B));
]- Output:
+ | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 * | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --- ------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 3 1 8 10 11 9 12 14 15 13 4 6 7 5 3 | 0 3 1 2 12 15 13 14 4 7 5 6 8 11 9 10 4 | 0 4 8 12 6 2 14 10 11 15 3 7 13 9 5 1 5 | 0 5 10 15 2 7 8 13 3 6 9 12 1 4 11 14 6 | 0 6 11 13 14 8 5 3 7 1 12 10 9 15 2 4 7 | 0 7 9 14 10 13 3 4 15 8 6 1 5 2 12 11 8 | 0 8 12 4 11 3 7 15 13 5 1 9 6 14 10 2 9 | 0 9 14 7 15 6 1 8 5 12 11 2 10 3 4 13 10 | 0 10 15 5 3 9 12 6 1 11 14 4 2 8 13 7 11 | 0 11 13 6 7 12 10 1 9 2 4 15 14 5 3 8 12 | 0 12 4 8 13 1 9 5 6 10 2 14 11 7 15 3 13 | 0 13 6 11 9 4 15 2 14 3 8 5 7 10 1 12 14 | 0 14 7 9 5 11 2 12 10 4 13 3 15 1 8 6 15 | 0 15 5 10 1 14 4 11 2 13 7 8 3 12 6 9 21508 + 42689 = 62149 21508 + 42689 = 35202