One Truth Only: Four Boxes, Four Inscriptions, One Medal
The maths fair at Nalanda School has a stall called 'One Truth Only'. Four identical steel boxes, numbered 1 to 4, sit on a table. Exactly one of them contains a brass medal; the other three are empty. On the lid of each box is engraved one sentence:
- Box 1: 'The medal is in box 2.'
- Box 2: 'The medal is not in box 1.'
- Box 3: 'The medal is in box 4.'
- Box 4: 'The medal is not in box 2.'
A card beside the stall reads: 'Exactly one of these four sentences is true. Name the box and win.' A visitor, Kabir, points out that the sentences on boxes 1 and 4 look like mirror images of each other and wonders whether that makes the puzzle unsolvable. The stall organiser says the card is not a trick, that it has a single correct answer, and that swapping 'one is true' for 'one is false' on the card would also give a single, but different, answer.
1. Using the card's rule 'exactly one sentence is true', find the medal. Show the full truth table.
Read the worked answer
Test each possible location and count the true sentences.
| Medal in | Box 1 | Box 2 | Box 3 | Box 4 | True count |
|---|---|---|---|---|---|
| Box 1 | F | F | F | T | 1 ✔ |
| Box 2 | T | T | F | F | 2 ✘ |
| Box 3 | F | T | F | T | 2 ✘ |
| Box 4 | F | T | T | T | 3 ✘ |
Only 'medal in box 1' gives exactly one true sentence (namely the one on box 4). So the medal is in box 1.
Sanity check: with the medal in box 1, box 1's claim is false, box 2's claim 'not in box 1' is false, box 3's claim is false, and box 4's claim 'not in box 2' is true. Exactly one true, as required.
2. Kabir worries that the mirrored sentences on boxes 1 and 4 make the puzzle unsolvable. Use the logic of the inscriptions to answer him without building the whole table.
Read the worked answer
Kabir has spotted something useful; he has simply drawn the wrong conclusion.
Box 1 says 'the medal is in box 2' and box 4 says 'the medal is not in box 2'. These are exact negations of each other, so exactly one of them is true and the other is false, whatever the medal's location.
But the card says only one sentence is true overall. That one true sentence is therefore already used up by boxes 1 and 4, which forces the remaining two sentences to be false:
- Box 2 says 'not in box 1'. False ⇒ the medal is in box 1.
- Box 3 says 'in box 4'. False ⇒ consistent, since the medal is in box 1.
So the medal is in box 1. Mirror-image sentences make a puzzle more constrained, not less; a statement and its negation always split one true and one false between them.
3. The organiser says the card could instead read 'exactly one sentence is false' and still have a single answer. What would that answer be?
Read the worked answer
'Exactly one false' means three sentences are true, which is the same as 'exactly one sentence is false'.
From the table in the first part, the true counts are 1, 2, 2 and 3 for the medal in boxes 1, 2, 3 and 4 respectively.
A count of 3 true sentences occurs only when the medal is in box 4.
Check directly: medal in box 4 makes box 1 false ('in box 2' is wrong), box 2 true, box 3 true and box 4 true - three truths and one falsehood, as required.
So flipping one word on the card moves the answer from box 1 to box 4. The two rules are genuinely different questions, which is why the exact wording on a logic card must be read with care.