Problems

At Willy Wonka’s chocolate factory, sweets are always packed into boxes of \(3\).

One day, Charlie, Veruca and Augustus each bring a bag of sweets to the factory. Charlie has some number of sweets, Veruca has one more sweet than Charlie, and Augustus has one more than Veruca.

When they put all their sweets together, will they be able to pack them perfectly into boxes of \(3\), with no sweets left over?

Friday shows Robinson Crusoe a magic trick:

He asks Robinson to write down any 15 whole numbers of his choice on a piece of paper. Then Friday looks at the list, and is always able to pick two of the numbers so that, when one is subtracted from the other, the result is a multiple of \(13\).

Can you explain why this trick works?

Alice was playing on the \(5\times 5\) lights out board and obtained this light pattern:

how did she obtain it?

After some playing with the \(3\times 3\) board, Sam guessed that there were \(900\) different light patterns that could be obtained by playing on this board. Was he right?

The original “Lights Out” game works like this: a light pattern is shown on the board, and your task is to turn all the lights off. A light pattern is called solvable if you can complete the game starting from that pattern. Ziheng and Jan are playing on an \(n\times n\) board, and they notice that some patterns are unsolvable. Can you find a rule to decide when a pattern is not solvable?

A rectangle has a perimeter of \(1\). Is it possible that its area is larger than \(1000\)?

Zahra has a \(3\times 3\) grid of little squares. Can she write the numbers \(2,4,6,7,8,10,12,14,16\) inside the little square - using each number exactly once - so that the sum of the three numbers in every row is the same?