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#38 Rotating dice

set by Christopher Dearlove

Put a cube on a surface and trace a square around its base. How many ways can you place the cube flat on the table and perfectly within the square so that its faces are oriented differently?

Imagine an axis going through the middle of two opposite faces of the cube. You can rotate it around this axis by 90 degrees, and the cube would take up the same space (but in a different orientation). How many different axes are there going through the cube where you can do this with some rotation less than 360 degrees?

For each of these axes, by which angles is it possible to rotate through and get back to a cube taking up the same space? Count the total number of possible rotation angles over all the axes (including the case of not rotating the cube, but only once).

Consider the number of ways to place the cube, and the total number of ways to rotate the cube, relative to the number of edges on it. Is this the same for other regular shapes used for dice? What other kinds of shape does this work for?

Solution next week

#37 Digital multiplication

Solution

Starting at 93, we get 93 ⇨ 27 ⇨ 14 ⇨ 4. To find the sequence of length 5, work back from 8, which comes from multiplying the digits of 18, 81, 24 or 42. To get 18, the prior number needs to be 29, 92, 36 or 63, as the non-trivial factors of 18 are 2, 9, 3 and 6. This stops when a number isn’t the product of two single digits, so rule out 29 (all primes are ruled out) and 92. To get 36, you need 49, 94 or 66, of which only 49 works; 63 could come from 79 or 97, but both are prime. Since 49 is the product of 7 × 7, the sequence goes 77 ⇨ 49 ⇨ 36 ⇨ 18 ⇨ 8. For three digits, we must avoid 0, or a 5 paired with an even number, as these end the sequence early. 688, 868 and 886 give 6-long sequences. Rearrangements of 679 also produce sequences of length 6, ending in 6.