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.
Michael made a cake, in the shape of a perfect cube, for 64 guests at a recent party.
The inside of the cake was sponge, and he iced the cake with red icing.
He didn't ice the bottom of the cake.
Michael cut each side of the cake into four equal pieces, making a total of 64 pieces of cake (each exactly the same size).
How many of the pieces of cake had at least 2 of their sides with icing?
Hint: How many pieces on the top layer have at least 2 sides with icing?
There were 12 pieces on the top layer with at least 2 sides with icing. The other three layers each have 4 pieces that have 2 sides with icing. A total of 24.