Henry Ernest Dudeney/Puzzles and Curious Problems/160 - Boxes of Cordite/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $160$
- Boxes of Cordite
- Cordite charges for $6$-inch howitzers were served out from ammunition dumps in boxes of $15$, $18$ and $20$.
- "Why the three different sizes of boxes?" I asked the officer on the dump.
- He answered: "So that we can give any battery the number of charges it needs without breaking a box.
- This was an excellent system for the delivery of a large number of boxes,
- but failed in small cases, like $5$, $10$, $25$ and $61$.
- Now, what is the biggest number of charges that cannot be served out in whole boxes of $15$, $18$ and $20$?
- It is not a very large number.
Solution
- $97$
Proof
The dump officer gives boxes of $18$ until the remainder is a multiple of $5$.
Then, unless this is $5$, $10$ or $25$, the remainder is given in $15$s or $20$s.
The biggest number for which this breaks down is $72 + 25 = 97$.
Take the case of a higher numbers, such as $133$.
$6$ boxes of $18$ makes $108$, leaving $25$.
So you give one box of $18$, leaving $115$, which can be delivered in one box of $15$ and five of $20$.
But in the case of $97$, $72$ is the first and only case leaving a multiple of $5$, that is, $25$.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $160$. -- Boxes of Cordite
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $227$. Boxes of Cordite