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#35 Cube cuts

set by Katie Steckles

Imagine a 3×3×3 cube (like a Rubik’s cube) that can be cut into separate pieces along the lines between the 27 individual small cubes that make up the larger shape.

If we are allowed to cut through more than one piece of the whole cube at once (as if passing a 2D plane through the whole shape), what is the fewest number of cuts we need to create four separate pieces of any size?

To cut the whole cube up into its 27 individual small cubes, we can make six cuts (two in each of the three directions). Convince yourself that this can’t be done in fewer than six, even if we are allowed to move pieces between cuts.

If, instead, you are only allowed to make cuts to one piece at a time – that is, taking one of the pieces each time and making a single cut that goes all the way through it – what is the fewest number of cuts you will need to separate the whole cube into its 27 individual cubes?

Solution next week

#34 Square dealing

Solution

After the famous 3² + 4² = 5² Pythagorean triple, the next sequence is 10² + 11² + 12² = 13² + 14², so the value of d is 10. The starting numbers follow the pattern 1×3, 2×5, 3×7, 4×9. They are all triangular numbers, skipping every second one (1+2, 1+2+3+4, 1+2+3+4+5+6).

The fifth sequence, of 11 numbers, starts with 5×11 = 55, that is: 55² + 56² + 57² + 58² + 59² + 60² = 61² + 62² + 63² + 64²+ 65².