BrainTwister #76: Tromino trials

#76 Tromino trials

Set by Colin Beveridge

There are four ways to tile a 5 × 3 rectangle with five 3 × 1 “trominoes”, as shown above.

If you have a 7 × 3 rectangle to tile with seven 3 × 1 trominoes, how many ways can the rectangle be tiled if exactly three are placed horizontally?

Other than three, what are the possible numbers of horizontal tiles in this rectangle?

In total, how many ways are there to tile the rectangle?

How many ways can you tile a 10 × 3 rectangle with trominoes?

Solution next week

#75 Letters and numbers

Solution

If TWO = 2, then T = 1 and W = 1. If THREE = 3, then also H = 1 and R = 1. If FOUR = 4, then F+R = 4 and thus F = 3. If FIVE = 5, then F+V = 5 and thus V = 2. If SIX = 6, then S+X = 6, but we can’t deduce either S or X yet. If SEVEN = 7, then S+V+N = 7 and thus S = 4, and, returning to SIX, X = 2. If EIGHT = 8, then G+H+T = 8 and thus G = 6. Then NINE can’t equal 9 as N = 1 and thus NINE = 2.

If ELEVEN = 11, then L+V+N = 11 and thus L = 8. Then TWELVE = 12, so it is self-describing. If TWENTY = 20, then T+W+N+T+Y = 20 and thus Y = 16.

All other numbers less than 20 aren’t self-describing.