Henry Ernest Dudeney/Puzzles and Curious Problems/38 - The Picnic/Solution/Proof 2

Proof
From:
 * $a + 2 b + 3 c + 4 d = 22$

we can infer that:
 * $a + 3 c = 22 - 2 b - 4 d$

is even.

Hence $a$ and $c$ must be of the same parity.

Suppose both $a$ and $c$ are even.

Then $b$ and $d$ are both odd.

Either $a = 2$ and $c = 4$ or vice versa.

In either case:


 * $a + 3 c \equiv 2 \pmod 4$

Because $b$ will be odd:
 * $2 p \equiv 2 \pmod 4$

and so:
 * $a + 3 c = 22 - 2 b - 4 d \equiv 0 \pmod 4$

Thus we have a contradiction.

So $a$ and $c$ must both be odd and therefore $b$ and $d$ must both be even.

$d$ cannot be $4$ since any assignment of $\set {1, 2, 3}$ to $\tuple {a, b, c}$ yields:
 * $a + 2 b + 3 c > 22 - \paren {4 \times 4} = 6$

So we must have $d = 2$ and $b = 4$, leading to:
 * $a + 3 c = 22 - \paren {2 \times 4} - \paren {4 \times 2} = 6$

from which $a = 3$ and $c = 1$ follow immediately.

So:
 * Jane drank the same quantity as her husband John MacGregor, that is, $3$ bottles
 * Lloyd Jones drank twice as much as the $4$ bottles drunk by his wife Elizabeth
 * William Smith drank three times as much as his wife Mary, who drank just $1$ bottle
 * Patrick Dolan drank four times as much as his wife Anne's $2$ bottles.