Here are the answers to the festive maths quiz I set on December 23. I hope you enjoyed it.

Read more: The magic of maths: festive puzzles to give your brain and imagination a workout

Puzzle 1: You are given nine gold coins that look identical. You are told that one of them is fake, and that this coin weighs less than the real ones. Using a set of old-fashioned balance scales, what is the smallest number of weighings you need to determine which is the fake coin?

Solution: You can do this in just two weighings:

(1) Divide the nine coins into three sets of three, and choose two of these sets to weigh against one another. If one set is lighter than the other, then the fake is one of these three coins. If the two sets weigh the same, then the fake is in the three unweighed coins.

(2) Now take the set with the fake coin, and weigh two of its coins against each other. If one is lighter, that is the fake. If they weigh the same, then the fake is the third coin.

Puzzle 2: You’ve been transported back in time to help cook Christmas dinner. Your job is to bake the Christmas pie, but all you’ve got is two egg-timers: one that times exactly four minutes, and one that times exactly seven minutes. How can you time ten minutes exactly?

Solution: There are multiple answers to this puzzle, but supposing the chef wants you cook this pie as quickly as possible, here’s how to do it:

- Start both timers at the same time.

- Once the four-minute timer has finished, the seven-minute timer will have three minutes to go. At this point, put the pie in the oven.

- Once the remaining three minutes on the seven-minute timer has finished, turn the seven-minute timer over.

- Let the seven-minute timer run its full course, then take the pie out immediately. The pie will have been in the oven for ten minutes exactly.

Puzzle 3: You are now entrusted with allocating the mulled wine, which is currently in two full ten-litre barrels. The chef hands you one five-litre bottle and one four-litre bottle, both empty. He orders you to fill the bottles with exactly three litres of wine each, without wasting a drop. How can you do this?

Solution: Here is a solution in 11 steps (see table below), recording the quantities of mulled wine in each barrel and bottle. B1 and B2 are the two ten-litre barrels; b5 and b4 are the five-litre and four-litre bottles respectively.

Note: You might have found a quicker solution than mine, but this is what I could come up with!

Puzzle 4: Suppose there are 100 days of Christmas. On the n-th day, you receive £n as a gift, ranging from £1 on the first day to £100 on the final day. Can you calculate the total amount of money you are given, without laboriously adding all 100 numbers........

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