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Hint: Try breaking each cog size into prime factors.

Answer: 30 revolutions.

If we break each wheel into its prime factors, we get:

63 = 3 x 3 x 7
42 = 2 x 3 x 7
35 = 5 x 7
27 = 3 x 3 x 3

We now think of rotating the large wheel just once, and this is 3 x 3 x 7 teeth moved (3 x 21), and we can see that 42 tooth wheel also has a 3 x 7 (21 teeth) in it, with an extra 2. If we therefore rotate the 63 toothed wheel twice, the 42 will have rotated three times.

The answer involves cancelling any common factors from the large wheel. We can cancel 3, 3, 7 from any of the smaller ones to leave 2 (from the 42), 5 (from the 35) and 3 (from the 27). 2 x 5 x 3 = 30. QED.

Puzzle 2

By changing the second letter of each word below, you can make another valid word.

Can you change each word such that the second letters will reveal an eleven letter word when read downwards.

Therefore, what now reads AWPYRNUCEPA will be a real word.

There are 1,609,344 millimetres in a mile and just over 1,259,842 sixteenths of inch in 2 kilometres.

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