# Candle puzzles – solutions

## Katie Steckles

Here are the solutions to the eight candle puzzles we posted earlier.

- The candles cost €1.05, and the matches cost €0.05.

- Every 30 seconds the number of people whose candles are lit doubles, starting with 1 person at time 0. After 30 seconds, 2 = 21 candles are lit, after 60 seconds 4 = 22 candles are lit, and so on. After 7 30-second gaps, 27 = 128 candles will be burning, so it will take until 8 lots of 30 seconds (4 minutes) have passed for all the candles to be lit.

- This can be achieved by lighting one of the candles at both ends, and allowing it to burn from both ends at the same time – the flames will meet in the middle after 30 minutes, at which point you can light the second candle and let it burn for an hour. (This does assume that the candles burn at the same rate regardless of which way up they are – in reality, if a candle was held upright and lit at both ends, the bottom end would burn more quickly. If you hold the candle horizontally it might also burn at a different rate than it burns from the top if held upright. But it’s a nice puzzle!)

- If you place three of the candles on each side of the scale, this will indicate which group of three the heavy candle is in: if the scale tips one way or the other, it’s in that group of three, and if it balances exactly, it’s in the three candles you didn’t use for this weighing. Take the group of three that the heavy candle is in, and place one candle on each side of the scale – then the same principle will apply to tell you which is the heavy one.

- After burning 100 candles you’ll have 10 candles’ worth of extra wax; this lets you make 10 more candles, each of which will yield another 1/10 of a candle of wax – so you could make 11 extra candles in total.

- If you blow on each candle once, starting with the one nearest to you and going round one at a time, when you get back to the start you’ll have blown out all the candles! This is because they each get blown on three times (three candles change from lit to unlit or vice versa on each blow), which is an odd number of times, so they’ll go from lit to unlit eventually.

- The number of candles you will have blown out is the sum of all the ages you’ve been – when you’re five you’ll have blown ou 1+2+3+4+5 = 15 candles. These are triangular numbers, and the general pattern is N×(N+1) / 2.

- If you have five 1s and five 0s, the first time you’ll hit a problem is when you need six of the same digit – which will happen for the first time when you try to write the number 111111 = 1+2+4+8+16+32 = 63.

Der Beitrag Candle puzzles – solutions erschien zuerst auf Heidelberg Laureate Forum.