The basic for this proposal is the classic circuit of adding two numbers on multiple bits.
Looking into the 2 dimensional array of the obtained appling the binary XOR operation on the bits of all the pairs of numbers smaller than a number it can be seen that instead of doing a+b (a and b are decimal numbres) we can do:
x= a xor b
y= 2*(a and b) and in approx. 50%, a+b = x or y and in the other 50% a+b = x+y this mean that we have to introduce again x and y in the ADD circuit.
If we draw the circuit only for adding 2 bits x and y to be introduced in the circuit means to apply some XOR gates to the binary representation of them.
The basic idea of this disclosure is focused on the 50% of cases in which a+b = x or y.
For this I formed a circuit in Tina ver. 6.0 a model of circuit that uses tri state buffers for selecting the cases when we have to use the XOR gates or the OR gates; instead of using a 3 state buffers there can be usesed one transistor.
In this way we avoid the usage of the XOR gates using only 1 transistors for each bit/line and OR gates.
In this way the timeof computing wll be faster and only the energy consume will be lower, beacuse of using less number of transistors.
ALU ADD Circuit, C Program source code
#include
#include
# include
# define MAX 32
FILE * pf;
int i, j;
int contor = 0;
int total = 0;
int rez;
void f(int x, int y)
{
fprintf(pf,"%3d + %3d = %3d + %3d",x,y, (x^y), 2*(x&y));
//fprintf(pf,"%3d ",x+y - ((x^y) | (2*(x&y))));
total ++;
if( ((x^y) + (2*(x&y))) == ((x^y) | (2*(x&y))) )
contor ++;
}
void main(void)
{
pf=fopen("c:\\xor.txt","w+");
for(i=0;i
fprintf(pf,"\n");
for(j=0;j
}
fcloseall();
printf("TOTAL = %d\n",total);
printf("OPTIMIZED = %d\n",contor);
printf("PERCENTAGE = %5.2f\n",(float)(contor)/(float)(total/100));
}
0 + 0 = 0 + 0 0 + 1 = 1 + 0 0 + 2 = 2 + 0 0 + 3 = 3 + 0 0 + 4 = 4 + 0 0 + 5 = 5 + 0 0 + 6 = 6 + 0 0 + 7 = 7 + 0 0 + 8 = 8 + 0 0 + 9 = 9 + 0 0 + 10 = 10 + 0 0 + 11 = 11 + 0 0 + 12 = 12 + 0 0 + 13 = 13 + 0 0 + 14 = 14 + 0 0 + 15 = 15 + 0 0 + 16 = 16 + 0 0 + 17 = 17 + 0 0 + 18 = 18 + 0 0 + 19 = 19 + 0 0 + 20 = 20 + 0 0 + 21 = 21 + 0 0 + 22 = 22 + 0 0 + 23 = 23 + 0 0 + 24 = 24 + 0 0 + 25 = 25 + 0 0 + 26 = 26 + 0 0 + 27 = 27 + 0 0 + 28 = 28 + 0 0 + 29 = 29 + 0 0 + 30 = 30 + 0 0 + 31 = 31 + 0
1 + 0 = 1 + 0 1 + 1 = 0 + 2 1 + 2 = 3 + 0 1 + 3 = 2 + 2 1 + 4 = 5 + 0 1 + 5 = 4 + 2 1 + 6 = 7 + 0 1 + 7 = 6 + 2 1 + 8 = 9 + 0 1 + 9 = 8 + 2 1 + 10 = 11 + 0 1 + 11 = 10 + 2 1 + 12 = 13 + 0 1 + 13 = 12 + 2 1 + 14 = 15 + 0 1 + 15 = 14 + 2 1 + 16 = 17 + 0 1 + 17 = 16 + 2 1 + 18 = 19 + 0 1 + 19 = 18 + 2 1 + 20 = 21 + 0 1 + 21 = 20 + 2 1 + 22 = 23 + 0 1 + 23 = 22 + 2 1 + 24 = 25 + 0 1 + 25 = 24 + 2 1 + 26 = 27 + 0 1 + 27 = 26 + 2 1 + 28 = 29 + 0 1 + 29 = 28 + 2 1 + 30 = 31 + 0 1 + 31 = 30 + 2
2 + 0 = 2 + 0 2 + 1 = 3 + 0 2 + 2 = 0 + 4 2 + 3 = 1 + 4 2 + 4 = 6 + 0 2 + 5 = 7 + 0 2 + 6 = 4 + 4 2 + 7 = 5 + 4 2 + 8 = 10 + 0 2 + 9 = 11 + 0 2 + 10 = 8 + 4 2 + 11 = 9 + 4 2 + 12 = 14 + 0 2 + 13 = 15 + 0 2 + 14 = 12 + 4 2 + 15 = 13 + 4 2 + 16 = 18 + 0 2 + 17 = 19 + 0 2 + 18 = 16 + 4 2 + 19 = 17 + 4 2 + 20 = 22 + 0 2 + 21 = 23 + 0 2 + 22 = 20 + 4 2 + 23 = 21 + 4 2 + 24 = 26 + 0 2 + 25 = 27 + 0 2 + 26 = 24 + 4 2 + 27 = 25 + 4 2 + 28 = 30 + 0 2 + 29 = 31 + 0 2 + 30 = 28 + 4 2 + 31 = 29 + 4
3 + 0 = 3 + 0 3 + 1 = 2 + 2 3 + 2 = 1 + 4 3 + 3 = 0 + 6 3 + 4 = 7 + 0 3 + 5 = 6 + 2 3 + 6 = 5 + 4 3 + 7 = 4 + 6 3 + 8 = 11 + 0 3 + 9 = 10 + 2 3 + 10 = 9 + 4 3 + 11 = 8 + 6 3 + 12 = 15 + 0 3 + 13 = 14 + 2 3 + 14 = 13 + 4 3 + 15 = 12 + 6 3 + 16 = 19 + 0 3 + 17 = 18 + 2 3 + 18 = 17 + 4 3 + 19 = 16 + 6 3 + 20 = 23 + 0 3 + 21 = 22 + 2 3 + 22 = 21 + 4 3 + 23 = 20 + 6 3 + 24 = 27 + 0 3 + 25 = 26 + 2 3 + 26 = 25 + 4 3 + 27 = 24 + 6 3 + 28 = 31 + 0 3 + 29 = 30 + 2 3 + 30 = 29 + 4 3 + 31 = 28 + 6
4 + 0 = 4 + 0 4 + 1 = 5 + 0 4 + 2 = 6 + 0 4 + 3 = 7 + 0 4 + 4 = 0 + 8 4 + 5 = 1 + 8 4 + 6 = 2 + 8 4 + 7 = 3 + 8 4 + 8 = 12 + 0 4 + 9 = 13 + 0 4 + 10 = 14 + 0 4 + 11 = 15 + 0 4 + 12 = 8 + 8 4 + 13 = 9 + 8 4 + 14 = 10 + 8 4 + 15 = 11 + 8 4 + 16 = 20 + 0 4 + 17 = 21 + 0 4 + 18 = 22 + 0 4 + 19 = 23 + 0 4 + 20 = 16 + 8 4 + 21 = 17 + 8 4 + 22 = 18 + 8 4 + 23 = 19 + 8 4 + 24 = 28 + 0 4 + 25 = 29 + 0 4 + 26 = 30 + 0 4 + 27 = 31 + 0 4 + 28 = 24 + 8 4 + 29 = 25 + 8 4 + 30 = 26 + 8 4 + 31 = 27 + 8
5 + 0 = 5 + 0 5 + 1 = 4 + 2 5 + 2 = 7 + 0 5 + 3 = 6 + 2 5 + 4 = 1 + 8 5 + 5 = 0 + 10 5 + 6 = 3 + 8 5 + 7 = 2 + 10 5 + 8 = 13 + 0 5 + 9 = 12 + 2 5 + 10 = 15 + 0 5 + 11 = 14 + 2 5 + 12 = 9 + 8 5 + 13 = 8 + 10 5 + 14 = 11 + 8 5 + 15 = 10 + 10 5 + 16 = 21 + 0 5 + 17 = 20 + 2 5 + 18 = 23 + 0 5 + 19 = 22 + 2 5 + 20 = 17 + 8 5 + 21 = 16 + 10 5 + 22 = 19 + 8 5 + 23 = 18 + 10 5 + 24 = 29 + 0 5 + 25 = 28 + 2 5 + 26 = 31 + 0 5 + 27 = 30 + 2 5 + 28 = 25 + 8 5 + 29 = 24 + 10 5 + 30 = 27 + 8 5 + 31 = 26 + 10
6 + 0 = 6 + 0 6 + 1 = 7 + 0 6 + 2 = 4 + 4 6 + 3 = 5 + 4 6 + 4 = 2 + 8 6 + 5 = 3 + 8 6 + 6 = 0 + 12 6 + 7 = 1 + 12 6 + 8 = 14 + 0 6 + 9 = 15 + 0 6 + 10 = 12 + 4 6 + 11 = 13 + 4 6 + 12 = 10 + 8 6 + 13 = 11 + 8 6 + 14 = 8 + 12 6 + 15 = 9 + 12 6 + 16 = 22 + 0 6 + 17 = 23 + 0 6 + 18 = 20 + 4 6 + 19 = 21 + 4 6 + 20 = 18 + 8 6 + 21 = 19 + 8 6 + 22 = 16 + 12 6 + 23 = 17 + 12 6 + 24 = 30 + 0 6 + 25 = 31 + 0 6 + 26 = 28 + 4 6 + 27 = 29 + 4 6 + 28 = 26 + 8 6 + 29 = 27 + 8 6 + 30 = 24 + 12 6 + 31 = 25 + 12
7 + 0 = 7 + 0 7 + 1 = 6 + 2 7 + 2 = 5 + 4 7 + 3 = 4 + 6 7 + 4 = 3 + 8 7 + 5 = 2 + 10 7 + 6 = 1 + 12 7 + 7 = 0 + 14 7 + 8 = 15 + 0 7 + 9 = 14 + 2 7 + 10 = 13 + 4 7 + 11 = 12 + 6 7 + 12 = 11 + 8 7 + 13 = 10 + 10 7 + 14 = 9 + 12 7 + 15 = 8 + 14 7 + 16 = 23 + 0 7 + 17 = 22 + 2 7 + 18 = 21 + 4 7 + 19 = 20 + 6 7 + 20 = 19 + 8 7 + 21 = 18 + 10 7 + 22 = 17 + 12 7 + 23 = 16 + 14 7 + 24 = 31 + 0 7 + 25 = 30 + 2 7 + 26 = 29 + 4 7 + 27 = 28 + 6 7 + 28 = 27 + 8 7 + 29 = 26 + 10 7 + 30 = 25 + 12 7 + 31 = 24 + 14
8 + 0 = 8 + 0 8 + 1 = 9 + 0 8 + 2 = 10 + 0 8 + 3 = 11 + 0 8 + 4 = 12 + 0 8 + 5 = 13 + 0 8 + 6 = 14 + 0 8 + 7 = 15 + 0 8 + 8 = 0 + 16 8 + 9 = 1 + 16 8 + 10 = 2 + 16 8 + 11 = 3 + 16 8 + 12 = 4 + 16 8 + 13 = 5 + 16 8 + 14 = 6 + 16 8 + 15 = 7 + 16 8 + 16 = 24 + 0 8 + 17 = 25 + 0 8 + 18 = 26 + 0 8 + 19 = 27 + 0 8 + 20 = 28 + 0 8 + 21 = 29 + 0 8 + 22 = 30 + 0 8 + 23 = 31 + 0 8 + 24 = 16 + 16 8 + 25 = 17 + 16 8 + 26 = 18 + 16 8 + 27 = 19 + 16 8 + 28 = 20 + 16 8 + 29 = 21 + 16 8 + 30 = 22 + 16 8 + 31 = 23 + 16
9 + 0 = 9 + 0 9 + 1 = 8 + 2 9 + 2 = 11 + 0 9 + 3 = 10 + 2 9 + 4 = 13 + 0 9 + 5 = 12 + 2 9 + 6 = 15 + 0 9 + 7 = 14 + 2 9 + 8 = 1 + 16 9 + 9 = 0 + 18 9 + 10 = 3 + 16 9 + 11 = 2 + 18 9 + 12 = 5 + 16 9 + 13 = 4 + 18 9 + 14 = 7 + 16 9 + 15 = 6 + 18 9 + 16 = 25 + 0 9 + 17 = 24 + 2 9 + 18 = 27 + 0 9 + 19 = 26 + 2 9 + 20 = 29 + 0 9 + 21 = 28 + 2 9 + 22 = 31 + 0 9 + 23 = 30 + 2 9 + 24 = 17 + 16 9 + 25 = 16 + 18 9 + 26 = 19 + 16 9 + 27 = 18 + 18 9 + 28 = 21 + 16 9 + 29 = 20 + 18 9 + 30 = 23 + 16 9 + 31 = 22 + 18
10 + 0 = 10 + 0 10 + 1 = 11 + 0 10 + 2 = 8 + 4 10 + 3 = 9 + 4 10 + 4 = 14 + 0 10 + 5 = 15 + 0 10 + 6 = 12 + 4 10 + 7 = 13 + 4 10 + 8 = 2 + 16 10 + 9 = 3 + 16 10 + 10 = 0 + 20 10 + 11 = 1 + 20 10 + 12 = 6 + 16 10 + 13 = 7 + 16 10 + 14 = 4 + 20 10 + 15 = 5 + 20 10 + 16 = 26 + 0 10 + 17 = 27 + 0 10 + 18 = 24 + 4 10 + 19 = 25 + 4 10 + 20 = 30 + 0 10 + 21 = 31 + 0 10 + 22 = 28 + 4 10 + 23 = 29 + 4 10 + 24 = 18 + 16 10 + 25 = 19 + 16 10 + 26 = 16 + 20 10 + 27 = 17 + 20 10 + 28 = 22 + 16 10 + 29 = 23 + 16 10 + 30 = 20 + 20 10 + 31 = 21 + 20
11 + 0 = 11 + 0 11 + 1 = 10 + 2 11 + 2 = 9 + 4 11 + 3 = 8 + 6 11 + 4 = 15 + 0 11 + 5 = 14 + 2 11 + 6 = 13 + 4 11 + 7 = 12 + 6 11 + 8 = 3 + 16 11 + 9 = 2 + 18 11 + 10 = 1 + 20 11 + 11 = 0 + 22 11 + 12 = 7 + 16 11 + 13 = 6 + 18 11 + 14 = 5 + 20 11 + 15 = 4 + 22 11 + 16 = 27 + 0 11 + 17 = 26 + 2 11 + 18 = 25 + 4 11 + 19 = 24 + 6 11 + 20 = 31 + 0 11 + 21 = 30 + 2 11 + 22 = 29 + 4 11 + 23 = 28 + 6 11 + 24 = 19 + 16 11 + 25 = 18 + 18 11 + 26 = 17 + 20 11 + 27 = 16 + 22 11 + 28 = 23 + 16 11 + 29 = 22 + 18 11 + 30 = 21 + 20 11 + 31 = 20 + 22
12 + 0 = 12 + 0 12 + 1 = 13 + 0 12 + 2 = 14 + 0 12 + 3 = 15 + 0 12 + 4 = 8 + 8 12 + 5 = 9 + 8 12 + 6 = 10 + 8 12 + 7 = 11 + 8 12 + 8 = 4 + 16 12 + 9 = 5 + 16 12 + 10 = 6 + 16 12 + 11 = 7 + 16 12 + 12 = 0 + 24 12 + 13 = 1 + 24 12 + 14 = 2 + 24 12 + 15 = 3 + 24 12 + 16 = 28 + 0 12 + 17 = 29 + 0 12 + 18 = 30 + 0 12 + 19 = 31 + 0 12 + 20 = 24 + 8 12 + 21 = 25 + 8 12 + 22 = 26 + 8 12 + 23 = 27 + 8 12 + 24 = 20 + 16 12 + 25 = 21 + 16 12 + 26 = 22 + 16 12 + 27 = 23 + 16 12 + 28 = 16 + 24 12 + 29 = 17 + 24 12 + 30 = 18 + 24 12 + 31 = 19 + 24
13 + 0 = 13 + 0 13 + 1 = 12 + 2 13 + 2 = 15 + 0 13 + 3 = 14 + 2 13 + 4 = 9 + 8 13 + 5 = 8 + 10 13 + 6 = 11 + 8 13 + 7 = 10 + 10 13 + 8 = 5 + 16 13 + 9 = 4 + 18 13 + 10 = 7 + 16 13 + 11 = 6 + 18 13 + 12 = 1 + 24 13 + 13 = 0 + 26 13 + 14 = 3 + 24 13 + 15 = 2 + 26 13 + 16 = 29 + 0 13 + 17 = 28 + 2 13 + 18 = 31 + 0 13 + 19 = 30 + 2 13 + 20 = 25 + 8 13 + 21 = 24 + 10 13 + 22 = 27 + 8 13 + 23 = 26 + 10 13 + 24 = 21 + 16 13 + 25 = 20 + 18 13 + 26 = 23 + 16 13 + 27 = 22 + 18 13 + 28 = 17 + 24 13 + 29 = 16 + 26 13 + 30 = 19 + 24 13 + 31 = 18 + 26
14 + 0 = 14 + 0 14 + 1 = 15 + 0 14 + 2 = 12 + 4 14 + 3 = 13 + 4 14 + 4 = 10 + 8 14 + 5 = 11 + 8 14 + 6 = 8 + 12 14 + 7 = 9 + 12 14 + 8 = 6 + 16 14 + 9 = 7 + 16 14 + 10 = 4 + 20 14 + 11 = 5 + 20 14 + 12 = 2 + 24 14 + 13 = 3 + 24 14 + 14 = 0 + 28 14 + 15 = 1 + 28 14 + 16 = 30 + 0 14 + 17 = 31 + 0 14 + 18 = 28 + 4 14 + 19 = 29 + 4 14 + 20 = 26 + 8 14 + 21 = 27 + 8 14 + 22 = 24 + 12 14 + 23 = 25 + 12 14 + 24 = 22 + 16 14 + 25 = 23 + 16 14 + 26 = 20 + 20 14 + 27 = 21 + 20 14 + 28 = 18 + 24 14 + 29 = 19 + 24 14 + 30 = 16 + 28 14 + 31 = 17 + 28
15 + 0 = 15 + 0 15 + 1 = 14 + 2 15 + 2 = 13 + 4 15 + 3 = 12 + 6 15 + 4 = 11 + 8 15 + 5 = 10 + 10 15 + 6 = 9 + 12 15 + 7 = 8 + 14 15 + 8 = 7 + 16 15 + 9 = 6 + 18 15 + 10 = 5 + 20 15 + 11 = 4 + 22 15 + 12 = 3 + 24 15 + 13 = 2 + 26 15 + 14 = 1 + 28 15 + 15 = 0 + 30 15 + 16 = 31 + 0 15 + 17 = 30 + 2 15 + 18 = 29 + 4 15 + 19 = 28 + 6 15 + 20 = 27 + 8 15 + 21 = 26 + 10 15 + 22 = 25 + 12 15 + 23 = 24 + 14 15 + 24 = 23 + 16 15 + 25 = 22 + 18 15 + 26 = 21 + 20 15 + 27 = 20 + 22 15 + 28 = 19 + 24 15 + 29 = 18 + 26 15 + 30 = 17 + 28 15 + 31 = 16 + 30
16 + 0 = 16 + 0 16 + 1 = 17 + 0 16 + 2 = 18 + 0 16 + 3 = 19 + 0 16 + 4 = 20 + 0 16 + 5 = 21 + 0 16 + 6 = 22 + 0 16 + 7 = 23 + 0 16 + 8 = 24 + 0 16 + 9 = 25 + 0 16 + 10 = 26 + 0 16 + 11 = 27 + 0 16 + 12 = 28 + 0 16 + 13 = 29 + 0 16 + 14 = 30 + 0 16 + 15 = 31 + 0 16 + 16 = 0 + 32 16 + 17 = 1 + 32 16 + 18 = 2 + 32 16 + 19 = 3 + 32 16 + 20 = 4 + 32 16 + 21 = 5 + 32 16 + 22 = 6 + 32 16 + 23 = 7 + 32 16 + 24 = 8 + 32 16 + 25 = 9 + 32 16 + 26 = 10 + 32 16 + 27 = 11 + 32 16 + 28 = 12 + 32 16 + 29 = 13 + 32 16 + 30 = 14 + 32 16 + 31 = 15 + 32
17 + 0 = 17 + 0 17 + 1 = 16 + 2 17 + 2 = 19 + 0 17 + 3 = 18 + 2 17 + 4 = 21 + 0 17 + 5 = 20 + 2 17 + 6 = 23 + 0 17 + 7 = 22 + 2 17 + 8 = 25 + 0 17 + 9 = 24 + 2 17 + 10 = 27 + 0 17 + 11 = 26 + 2 17 + 12 = 29 + 0 17 + 13 = 28 + 2 17 + 14 = 31 + 0 17 + 15 = 30 + 2 17 + 16 = 1 + 32 17 + 17 = 0 + 34 17 + 18 = 3 + 32 17 + 19 = 2 + 34 17 + 20 = 5 + 32 17 + 21 = 4 + 34 17 + 22 = 7 + 32 17 + 23 = 6 + 34 17 + 24 = 9 + 32 17 + 25 = 8 + 34 17 + 26 = 11 + 32 17 + 27 = 10 + 34 17 + 28 = 13 + 32 17 + 29 = 12 + 34 17 + 30 = 15 + 32 17 + 31 = 14 + 34
18 + 0 = 18 + 0 18 + 1 = 19 + 0 18 + 2 = 16 + 4 18 + 3 = 17 + 4 18 + 4 = 22 + 0 18 + 5 = 23 + 0 18 + 6 = 20 + 4 18 + 7 = 21 + 4 18 + 8 = 26 + 0 18 + 9 = 27 + 0 18 + 10 = 24 + 4 18 + 11 = 25 + 4 18 + 12 = 30 + 0 18 + 13 = 31 + 0 18 + 14 = 28 + 4 18 + 15 = 29 + 4 18 + 16 = 2 + 32 18 + 17 = 3 + 32 18 + 18 = 0 + 36 18 + 19 = 1 + 36 18 + 20 = 6 + 32 18 + 21 = 7 + 32 18 + 22 = 4 + 36 18 + 23 = 5 + 36 18 + 24 = 10 + 32 18 + 25 = 11 + 32 18 + 26 = 8 + 36 18 + 27 = 9 + 36 18 + 28 = 14 + 32 18 + 29 = 15 + 32 18 + 30 = 12 + 36 18 + 31 = 13 + 36
19 + 0 = 19 + 0 19 + 1 = 18 + 2 19 + 2 = 17 + 4 19 + 3 = 16 + 6 19 + 4 = 23 + 0 19 + 5 = 22 + 2 19 + 6 = 21 + 4 19 + 7 = 20 + 6 19 + 8 = 27 + 0 19 + 9 = 26 + 2 19 + 10 = 25 + 4 19 + 11 = 24 + 6 19 + 12 = 31 + 0 19 + 13 = 30 + 2 19 + 14 = 29 + 4 19 + 15 = 28 + 6 19 + 16 = 3 + 32 19 + 17 = 2 + 34 19 + 18 = 1 + 36 19 + 19 = 0 + 38 19 + 20 = 7 + 32 19 + 21 = 6 + 34 19 + 22 = 5 + 36 19 + 23 = 4 + 38 19 + 24 = 11 + 32 19 + 25 = 10 + 34 19 + 26 = 9 + 36 19 + 27 = 8 + 38 19 + 28 = 15 + 32 19 + 29 = 14 + 34 19 + 30 = 13 + 36 19 + 31 = 12 + 38
20 + 0 = 20 + 0 20 + 1 = 21 + 0 20 + 2 = 22 + 0 20 + 3 = 23 + 0 20 + 4 = 16 + 8 20 + 5 = 17 + 8 20 + 6 = 18 + 8 20 + 7 = 19 + 8 20 + 8 = 28 + 0 20 + 9 = 29 + 0 20 + 10 = 30 + 0 20 + 11 = 31 + 0 20 + 12 = 24 + 8 20 + 13 = 25 + 8 20 + 14 = 26 + 8 20 + 15 = 27 + 8 20 + 16 = 4 + 32 20 + 17 = 5 + 32 20 + 18 = 6 + 32 20 + 19 = 7 + 32 20 + 20 = 0 + 40 20 + 21 = 1 + 40 20 + 22 = 2 + 40 20 + 23 = 3 + 40 20 + 24 = 12 + 32 20 + 25 = 13 + 32 20 + 26 = 14 + 32 20 + 27 = 15 + 32 20 + 28 = 8 + 40 20 + 29 = 9 + 40 20 + 30 = 10 + 40 20 + 31 = 11 + 40
21 + 0 = 21 + 0 21 + 1 = 20 + 2 21 + 2 = 23 + 0 21 + 3 = 22 + 2 21 + 4 = 17 + 8 21 + 5 = 16 + 10 21 + 6 = 19 + 8 21 + 7 = 18 + 10 21 + 8 = 29 + 0 21 + 9 = 28 + 2 21 + 10 = 31 + 0 21 + 11 = 30 + 2 21 + 12 = 25 + 8 21 + 13 = 24 + 10 21 + 14 = 27 + 8 21 + 15 = 26 + 10 21 + 16 = 5 + 32 21 + 17 = 4 + 34 21 + 18 = 7 + 32 21 + 19 = 6 + 34 21 + 20 = 1 + 40 21 + 21 = 0 + 42 21 + 22 = 3 + 40 21 + 23 = 2 + 42 21 + 24 = 13 + 32 21 + 25 = 12 + 34 21 + 26 = 15 + 32 21 + 27 = 14 + 34 21 + 28 = 9 + 40 21 + 29 = 8 + 42 21 + 30 = 11 + 40 21 + 31 = 10 + 42
22 + 0 = 22 + 0 22 + 1 = 23 + 0 22 + 2 = 20 + 4 22 + 3 = 21 + 4 22 + 4 = 18 + 8 22 + 5 = 19 + 8 22 + 6 = 16 + 12 22 + 7 = 17 + 12 22 + 8 = 30 + 0 22 + 9 = 31 + 0 22 + 10 = 28 + 4 22 + 11 = 29 + 4 22 + 12 = 26 + 8 22 + 13 = 27 + 8 22 + 14 = 24 + 12 22 + 15 = 25 + 12 22 + 16 = 6 + 32 22 + 17 = 7 + 32 22 + 18 = 4 + 36 22 + 19 = 5 + 36 22 + 20 = 2 + 40 22 + 21 = 3 + 40 22 + 22 = 0 + 44 22 + 23 = 1 + 44 22 + 24 = 14 + 32 22 + 25 = 15 + 32 22 + 26 = 12 + 36 22 + 27 = 13 + 36 22 + 28 = 10 + 40 22 + 29 = 11 + 40 22 + 30 = 8 + 44 22 + 31 = 9 + 44
23 + 0 = 23 + 0 23 + 1 = 22 + 2 23 + 2 = 21 + 4 23 + 3 = 20 + 6 23 + 4 = 19 + 8 23 + 5 = 18 + 10 23 + 6 = 17 + 12 23 + 7 = 16 + 14 23 + 8 = 31 + 0 23 + 9 = 30 + 2 23 + 10 = 29 + 4 23 + 11 = 28 + 6 23 + 12 = 27 + 8 23 + 13 = 26 + 10 23 + 14 = 25 + 12 23 + 15 = 24 + 14 23 + 16 = 7 + 32 23 + 17 = 6 + 34 23 + 18 = 5 + 36 23 + 19 = 4 + 38 23 + 20 = 3 + 40 23 + 21 = 2 + 42 23 + 22 = 1 + 44 23 + 23 = 0 + 46 23 + 24 = 15 + 32 23 + 25 = 14 + 34 23 + 26 = 13 + 36 23 + 27 = 12 + 38 23 + 28 = 11 + 40 23 + 29 = 10 + 42 23 + 30 = 9 + 44 23 + 31 = 8 + 46
24 + 0 = 24 + 0 24 + 1 = 25 + 0 24 + 2 = 26 + 0 24 + 3 = 27 + 0 24 + 4 = 28 + 0 24 + 5 = 29 + 0 24 + 6 = 30 + 0 24 + 7 = 31 + 0 24 + 8 = 16 + 16 24 + 9 = 17 + 16 24 + 10 = 18 + 16 24 + 11 = 19 + 16 24 + 12 = 20 + 16 24 + 13 = 21 + 16 24 + 14 = 22 + 16 24 + 15 = 23 + 16 24 + 16 = 8 + 32 24 + 17 = 9 + 32 24 + 18 = 10 + 32 24 + 19 = 11 + 32 24 + 20 = 12 + 32 24 + 21 = 13 + 32 24 + 22 = 14 + 32 24 + 23 = 15 + 32 24 + 24 = 0 + 48 24 + 25 = 1 + 48 24 + 26 = 2 + 48 24 + 27 = 3 + 48 24 + 28 = 4 + 48 24 + 29 = 5 + 48 24 + 30 = 6 + 48 24 + 31 = 7 + 48
25 + 0 = 25 + 0 25 + 1 = 24 + 2 25 + 2 = 27 + 0 25 + 3 = 26 + 2 25 + 4 = 29 + 0 25 + 5 = 28 + 2 25 + 6 = 31 + 0 25 + 7 = 30 + 2 25 + 8 = 17 + 16 25 + 9 = 16 + 18 25 + 10 = 19 + 16 25 + 11 = 18 + 18 25 + 12 = 21 + 16 25 + 13 = 20 + 18 25 + 14 = 23 + 16 25 + 15 = 22 + 18 25 + 16 = 9 + 32 25 + 17 = 8 + 34 25 + 18 = 11 + 32 25 + 19 = 10 + 34 25 + 20 = 13 + 32 25 + 21 = 12 + 34 25 + 22 = 15 + 32 25 + 23 = 14 + 34 25 + 24 = 1 + 48 25 + 25 = 0 + 50 25 + 26 = 3 + 48 25 + 27 = 2 + 50 25 + 28 = 5 + 48 25 + 29 = 4 + 50 25 + 30 = 7 + 48 25 + 31 = 6 + 50
26 + 0 = 26 + 0 26 + 1 = 27 + 0 26 + 2 = 24 + 4 26 + 3 = 25 + 4 26 + 4 = 30 + 0 26 + 5 = 31 + 0 26 + 6 = 28 + 4 26 + 7 = 29 + 4 26 + 8 = 18 + 16 26 + 9 = 19 + 16 26 + 10 = 16 + 20 26 + 11 = 17 + 20 26 + 12 = 22 + 16 26 + 13 = 23 + 16 26 + 14 = 20 + 20 26 + 15 = 21 + 20 26 + 16 = 10 + 32 26 + 17 = 11 + 32 26 + 18 = 8 + 36 26 + 19 = 9 + 36 26 + 20 = 14 + 32 26 + 21 = 15 + 32 26 + 22 = 12 + 36 26 + 23 = 13 + 36 26 + 24 = 2 + 48 26 + 25 = 3 + 48 26 + 26 = 0 + 52 26 + 27 = 1 + 52 26 + 28 = 6 + 48 26 + 29 = 7 + 48 26 + 30 = 4 + 52 26 + 31 = 5 + 52
27 + 0 = 27 + 0 27 + 1 = 26 + 2 27 + 2 = 25 + 4 27 + 3 = 24 + 6 27 + 4 = 31 + 0 27 + 5 = 30 + 2 27 + 6 = 29 + 4 27 + 7 = 28 + 6 27 + 8 = 19 + 16 27 + 9 = 18 + 18 27 + 10 = 17 + 20 27 + 11 = 16 + 22 27 + 12 = 23 + 16 27 + 13 = 22 + 18 27 + 14 = 21 + 20 27 + 15 = 20 + 22 27 + 16 = 11 + 32 27 + 17 = 10 + 34 27 + 18 = 9 + 36 27 + 19 = 8 + 38 27 + 20 = 15 + 32 27 + 21 = 14 + 34 27 + 22 = 13 + 36 27 + 23 = 12 + 38 27 + 24 = 3 + 48 27 + 25 = 2 + 50 27 + 26 = 1 + 52 27 + 27 = 0 + 54 27 + 28 = 7 + 48 27 + 29 = 6 + 50 27 + 30 = 5 + 52 27 + 31 = 4 + 54
28 + 0 = 28 + 0 28 + 1 = 29 + 0 28 + 2 = 30 + 0 28 + 3 = 31 + 0 28 + 4 = 24 + 8 28 + 5 = 25 + 8 28 + 6 = 26 + 8 28 + 7 = 27 + 8 28 + 8 = 20 + 16 28 + 9 = 21 + 16 28 + 10 = 22 + 16 28 + 11 = 23 + 16 28 + 12 = 16 + 24 28 + 13 = 17 + 24 28 + 14 = 18 + 24 28 + 15 = 19 + 24 28 + 16 = 12 + 32 28 + 17 = 13 + 32 28 + 18 = 14 + 32 28 + 19 = 15 + 32 28 + 20 = 8 + 40 28 + 21 = 9 + 40 28 + 22 = 10 + 40 28 + 23 = 11 + 40 28 + 24 = 4 + 48 28 + 25 = 5 + 48 28 + 26 = 6 + 48 28 + 27 = 7 + 48 28 + 28 = 0 + 56 28 + 29 = 1 + 56 28 + 30 = 2 + 56 28 + 31 = 3 + 56
29 + 0 = 29 + 0 29 + 1 = 28 + 2 29 + 2 = 31 + 0 29 + 3 = 30 + 2 29 + 4 = 25 + 8 29 + 5 = 24 + 10 29 + 6 = 27 + 8 29 + 7 = 26 + 10 29 + 8 = 21 + 16 29 + 9 = 20 + 18 29 + 10 = 23 + 16 29 + 11 = 22 + 18 29 + 12 = 17 + 24 29 + 13 = 16 + 26 29 + 14 = 19 + 24 29 + 15 = 18 + 26 29 + 16 = 13 + 32 29 + 17 = 12 + 34 29 + 18 = 15 + 32 29 + 19 = 14 + 34 29 + 20 = 9 + 40 29 + 21 = 8 + 42 29 + 22 = 11 + 40 29 + 23 = 10 + 42 29 + 24 = 5 + 48 29 + 25 = 4 + 50 29 + 26 = 7 + 48 29 + 27 = 6 + 50 29 + 28 = 1 + 56 29 + 29 = 0 + 58 29 + 30 = 3 + 56 29 + 31 = 2 + 58
30 + 0 = 30 + 0 30 + 1 = 31 + 0 30 + 2 = 28 + 4 30 + 3 = 29 + 4 30 + 4 = 26 + 8 30 + 5 = 27 + 8 30 + 6 = 24 + 12 30 + 7 = 25 + 12 30 + 8 = 22 + 16 30 + 9 = 23 + 16 30 + 10 = 20 + 20 30 + 11 = 21 + 20 30 + 12 = 18 + 24 30 + 13 = 19 + 24 30 + 14 = 16 + 28 30 + 15 = 17 + 28 30 + 16 = 14 + 32 30 + 17 = 15 + 32 30 + 18 = 12 + 36 30 + 19 = 13 + 36 30 + 20 = 10 + 40 30 + 21 = 11 + 40 30 + 22 = 8 + 44 30 + 23 = 9 + 44 30 + 24 = 6 + 48 30 + 25 = 7 + 48 30 + 26 = 4 + 52 30 + 27 = 5 + 52 30 + 28 = 2 + 56 30 + 29 = 3 + 56 30 + 30 = 0 + 60 30 + 31 = 1 + 60
31 + 0 = 31 + 0 31 + 1 = 30 + 2 31 + 2 = 29 + 4 31 + 3 = 28 + 6 31 + 4 = 27 + 8 31 + 5 = 26 + 10 31 + 6 = 25 + 12 31 + 7 = 24 + 14 31 + 8 = 23 + 16 31 + 9 = 22 + 18 31 + 10 = 21 + 20 31 + 11 = 20 + 22 31 + 12 = 19 + 24 31 + 13 = 18 + 26 31 + 14 = 17 + 28 31 + 15 = 16 + 30 31 + 16 = 15 + 32 31 + 17 = 14 + 34 31 + 18 = 13 + 36 31 + 19 = 12 + 38 31 + 20 = 11 + 40 31 + 21 = 10 + 42 31 + 22 = 9 + 44 31 + 23 = 8 + 46 31 + 24 = 7 + 48 31 + 25 = 6 + 50 31 + 26 = 5 + 52 31 + 27 = 4 + 54 31 + 28 = 3 + 56 31 + 29 = 2 + 58 31 + 30 = 1 + 60 31 + 31 = 0 + 62
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