Henry Ernest Dudeney puzzle

Question

An officer explained that the force to which he belonged originally consisted of 1000 men, but that it lost heavily in an engagement, and the survivors surrendered and were marched down to a concentration camp.

• On the first day's march one-sixth of the survivors escaped;
• On the second day one-eighth of the remainder escaped, and one man died;
• When they arrived in camp, the rest were set to work in four equal gangs.

How many had been killed in the engagement?

Answer 1

Solving the puzzle as written:

Start from the bottom.

x = number of people per gang (positive integer)

4x = number of people who arrived in camp

(4x + 1) * 8/7 = number of people at start of day 2

(4x + 1) * 8/7 * 6/5 = number of people at start of day 1 (less than 1000)

4x + 1 is a multiple of 35

Inspection of small multiples gives x = 26 + 35y for some non-negative integer y

Thus, (4x + 1) * 8/7 * 6/5 = 144 + 192y

y is at most 4, so number of people at start of day 1 is 144, 336, 528, 720, or 912

Now if Jaap's piece is added:

z = number of people per gang (positive integer)

4z = number of people who arrived in camp

4z * 4/3 = number of people at start of day 3

(16z/3 + 1) * 8/7 = number of people at start of day 2

(16z/3 + 1) * 8/7 * 6/5 = number of people at start of day 1 (less than 1000)

Let 4x = 16z/3, i.e. x = 4z/3, so the solutions are a subset of the above

x (previously 26, 61, 96, 131, or 166) is a multiple of 3, so it must be 96, and the number of people at start of day 1 is 528