The numbers 1 to 5 are on a board. Rub out any two and write their sum minus one. Keep going until one number is left. Every child in the room chooses differently and every one of them ends on 11.
The question is not what the answer is. It is why the order cannot matter, and that is a different kind of question from every other challenge sheet here. The others hand over facts that pin an answer down; this one hands over a process with a completely free choice at every step, and the answer is the same anyway.
Why this sheet works
The argument fits in two observations. Each move rubs out two numbers and writes one, so the count falls by exactly one — five numbers means four moves, always. Each move also drops the total by exactly one. Four moves, four lost, and the numbers added to 15 to start with: the last number was 11 before anybody touched the board.
Find what every move leaves alone. Show why the order cannot matter.
1The numbers 3 to 13 are written on a board. Rub out any two of them and write their sum plus 2 in their place. Keep going until one is left.What is the last number?Why
2The numbers 1 to 10 are written on cards. Two cards are taken away and replaced by one showing their sum plus 3. Repeat until one card is left.What number is on the last card?Why
Invariant puzzlesUp to 16 numbers
Name: Date:
Find what every move leaves alone. Show why the order cannot matter.
3The numbers 4 to 19 are written on a board. Rub out any two of them and write their sum minus 1 in their place. Keep going until one is left.What is the last number?Why
4The numbers 3 to 11 are written on a board. Rub out any two of them and write their sum minus 3 in their place. Keep going until one is left.What is the last number?Why
Invariant puzzles · Up to 16 numbersAnswer key
1. Each move rubs out two numbers and writes one in their place, so the count falls by 1 each time: 11 numbers means 10 moves. Each move also raises the TOTAL by exactly 2, so 10 × 2 = 20 is gained altogether. The numbers add to 88, so whatever order anyone uses, the last number is 88 + 20 = 108.2. Each move takes away two cards and puts one back, so the count falls by 1 each time: 10 numbers means 9 moves. Each move also raises the TOTAL by exactly 3, so 9 × 3 = 27 is gained altogether. The numbers add to 55, so whatever order anyone uses, the number on the last card is 55 + 27 = 82.3. Each move rubs out two numbers and writes one in their place, so the count falls by 1 each time: 16 numbers means 15 moves. Each move also drops the TOTAL by exactly 1, so 15 × 1 = 15 is lost altogether. The numbers add to 184, so whatever order anyone uses, the last number is 184 − 15 = 169.4. Each move rubs out two numbers and writes one in their place, so the count falls by 1 each time: 9 numbers means 8 moves. Each move also drops the TOTAL by exactly 3, so 8 × 3 = 24 is lost altogether. The numbers add to 63, so whatever order anyone uses, the last number is 63 − 24 = 39.
