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#229 The greater gouda

Set by Brian Hobbs

Madeline has many large cheese cubes to cut into smaller cubes for tonight’s charity event. She wants to turn some of the large cubes into 4x4x4 smaller ones, creating 64 in total (see image A). She could do that with nine straight cuts, three along each axis, but her knife is as dull as the task, and she realises that if she rearranges the pieces between cuts, she can double their number with every cut she makes, and thus get the same result with fewer slices. After the first cut, there are two pieces; after the second, four; after the third, eight, and so on, giving 64 cubes with only six cuts.

Likewise, a 3x3x3 stack of cubes (image B), or 27 total cubes, could be achieved in six cuts without moving any pieces around, but she thinks she should be able to, well, cut that number down if she similarly rearranges the pieces between each slice. What is the minimum number of cuts she will need to end up with 27 cheese cubes?

Solution next week

#228 A knight on the tiles

Solution

The three dots specify the locations of nine diagonals, but only touch eight squares. Therefore, two of them – the two black dots – must share a diagonal. The other three squares neighbouring the white dot must have white diagonals, and then the pairs of tiles remaining next to the black dots must have black diagonals. The top-right tile must be placed to prevent an island forming.