Henry Ernest Dudeney/Puzzles and Curious Problems/176 - Counting the Loss/Solution

by : $176$

 * Counting the Loss

Solution
There were $472$ soldiers who were killed in the engagement.

Proof
Let $n$ be the number of survivors of the engagement.

Let $n_1$, $n_2$ and $n_3$ be the numbers left in the captured group at the end of days $1$ to $3$ respectively.

We have:

This leads us to the linear Diophantine equation:
 * $35 n - 256 k = 48$

From Example of Linear Diophantine Equation: $35 x - 256 y = 48$:
 * $n = 16 + 256 t, k = 2 + 35 t$

for $t \in \Z$.

We now need to select a value of $t$ such that:
 * $1 \le n \le 1000$

and:
 * $6 \divides n$

in order to fit the conditions of the question.

By inspection of the cases where $t = 0, 1, 2, 3, 4$ we have that the only solution is for $t = 2$.

Thus:
 * $n = 16 + 256 \times 2 = 528$

That is, there were $528$ survivors of the initial combat.

Hence $1000 - 528 = 472$ men died in battle.

The result follows.