## Two candles brainteaser

By David Shaw, 27 July 2022

**Difficulty: Extreme!**

Barry has 2 candles that are both 60 centimetres long. One will last for 1 hour, and the other will last for 2 hours.

Barry lights both candles at the same time. Later, he blows them out at the same time.

The next day, Barry realises that one of the candles is half the length of the other.

Assuming they both burnt evenly, how long were they burning for?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

After 1 hour, the fast-burning candle would be gone. As it still has some length, the candles were burning for less than an hour.

Since the fast candle ends up half the height of the slow one, it’s going to be more than half burnt. So the candles would have been burning for more than 30 minutes.

## Brainteaser answer

One way to solve this puzzle is simply to try some numbers and see what comes out. But a bit of clever thinking can help.

One way to imagine the candles is that they are made of burnt and unburnt sections. As they burn, the unburnt section is slowly transformed into burnt. But throughout the process, the total length of burnt plus unburnt stays the same for both candles.

The fast candle has a burnt section that’s twice as long as the unburnt section of the slow candle. Meanwhile, at the end of the puzzle, the unburnt section of the slow candle is twice the length of the unburnt section of the fast candle.

There’s a matching pattern here. Each candle has a short section and a long section. Let’s take a guess and say that the (short) burnt section of the slow candle is the same length as the (short) unburnt part of the fast candle.

Then the slow candle is ⅓ burnt, and ⅔ unburnt, and the fast candle is ⅔ burnt and ⅓ unburnt. It takes ⅔ of an hour (or 40 minutes) for the fast candle to burn that far, so that’s our answer!

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5 August, 2022 at 3:56 pm

Can also create mathematical equations to solve this.

Let the length of the fast candle burnt be y centimetres and the length of the slow candle burnt be x centimetres.

The remaining length of both candles in centimetres after being burnt has the following relationship:

60 – y = 0.5(60 – x)

This can be simplified to:

x = 2y – 60

Second relationship between y and x is that the fast candle burns at the rate of 1 centimetre per 1 minute since 60 cm = 1 hour or 60 cm = 60 minutes.

The slow candle burns at 60 cm = 2 hours, or 30 cm per 60 minutes, so 0.5 cm per 1 minute.

Therefore the relationship between y and x for 1 minute is: y = 2x.

Substituting the second equation into the first:

x = 2(2x) – 60

x = 4x – 60

-3x = -60

so x = 20 cm.

Since we know that for every 0.5 cm burned in the slow candle, 1 minute lapses, then for 20 cm burned, (20 x 2) = 40 minutes has lapsed.

Resulting lengths of the slow candle are 60 – 20 = 40 cm, and fast candle is 20 cm (= 60 – 40) after being burnt.

5 August, 2022 at 6:39 pm

It doesn’t say in the puzzle that it is the fast candle that is half the length of the slow candle. It just says that “one of the candles is half the length of the other”. Does it make a difference which one is smaller?

6 August, 2022 at 1:18 pm

Hi Kate,

They both start at the same length and the fast one burns down more quickly. And that means the fast one will always be shorter than the slow one.

If you try to solve the puzzle assuming the fast candle is longer than the slow one, I think you get negative infinity burning time, and final lengths of infinity for both candles!