Bit-shifting

grains.h

#ifndef GRAINS_H
#define GRAINS_H

#include <stdint.h>

uint64_t square(uint8_t index);
uint64_t total(void);

#endif

grains.c

#include "grains.h"

uint64_t square(uint8_t index)
{
    return (index > 0 && index < 65) ? 1ul << (index - 1) : 0;
}

uint64_t total(void)
{
    return ((((uint64_t)1 << 63) - 1) << 1) + 1;
}

Instead of using math to calculate the number of grains on a square, you can set a bit in the correct position of a uint64_t.

To understand how this works, consider just two squares that are represented in binary bits as 00.

You use the left-shift operator (<<) to set 1 at the position needed to make the correct decimal value.

• To set the one grain on Square One you shift 1 for 0 positions to the left. So, if index is 1 for square One, you subtract index by 1 to get 0, which will not move it any positions to the left. The result is binary 01, which is decimal 1.
• To set the two grains on Square Two you shift 1 for 1 position to the left. So, if index is 2 for square Two, you subtract index by 1 to get 1, which will move it 1 position to the left. The result is binary 10, which is decimal 2.

For total we want all of the 64 bits set to 1 to get the sum of grains on all sixty-four squares. The easy way to do this is to set the 65th bit to 1 and then subtract 1. However, we can't do this with a uint64_t which has only 64 bits. Using a four squares example, if the number were only four bits long, we can't set the fifth bit.

• We can shift 1 left three times to get 1000 binary.
• We can then subtract 1 from it to get a number of all 1s except for the leftmost bit (0111).
• We can then shift that number once to the left to get all 1s except for the righttmost bit 1110.
• Finally, we can add 1 to that number to get all 1s (1111).

The same can be done with 64 squares by shifting 1 left 63 times.

• We can then subtract 1 from it to get a number of all 1s except for the leftmost bit.
• We can then shift that number once to the left to get all 1s except for the righttmost bit.
• Finally, we can add 1 to that number to get all 1s.

Optimization

#include "grains.h"
#include <limits.h>

uint64_t total(void)
{
    return UINT64_MAX;
}

The UINT64_MAX value from the limits.h file is the value of having all 64 bits in a uint64_t set to 1. That is the value needed to represent the doubling of grains from the first square through the 64th square.

For example, four squares would be represented by 1111 which is 0001 + 0010 + 0100 + 1000 in binary, or 1 + 2 + 4 + 8 in decimal. Binary 1111 is 15 in decimal, which would be the maximum value of a four-bit unsigned integer (0 through 15).