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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 of course did not 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?

[Ref: ZPJN]

Hint: How many pieces on the top layer have at least 2 sides with icing?

Answer: 24.

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.

Puzzle 397

Place letters into the grid such that every row, column, and 2x2 block has letters (in any order) that form a common word.

Each letter is only used once, and no letter is repeated in the rows/cols/blocks.

Across: DUCK, PLUM, TOES, BALE
Down : DOME, PLUS, CUBE, TALK
Boxes : DUMP, LUCK, SOLE, BEAT
Note : Other anagrams of these words are OK as long as they don't change the answer grid.

Puzzle 398

I have a machine which has four sequential cog wheels in constant mesh.

The largest cog has 168 teeth and the others have 49, 32 and 15 respectively.

How many revolutions must the largest cog make before each of the cogs is back in its starting position?

Hint: Try breaking each cog size into prime factors.

Answer: 140 revolutions.

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

168= 2 x 2 x 2 x 3 x 7
49 = 7 x 7
32 = 2 x 2 x 2 x 2 x 2
15 = 3 x 5

So

revolutions x 168
---------------------------------
2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 7

revolutions x 2 x 2 x 2 x 3 x 7
---------------------------------
2 x 2 x 2 x 2 x 2 x 3 x 5 x 7 x 7

revolutions
-------------
2 x 2 x 5 x 7

revolutions
-------------
140

Which means revolutions = 140. QED.

Puzzle 399

Starting with the 6 in the bottom left corner, what is the highest total you can make, if you only move up or right, using the mathematical signs on the way?

Using the BrainTracker grid below, how many words can you find? Each word must contain the central K and no letter can be used twice, however, the letters do not have to be connected. Proper nouns are not allowed, however, plurals are. Can you find the nine letter word?

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