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#32 Crossed out of order

set by Paul Taylor

If we write the numbers 1 to 4 down in the order 3, 1, 4, 2 and then cross out the 1, the remaining sequence starts with the middle number out of the ones that are left, followed by the largest and then the smallest. (If we had crossed out a different number instead, this wouldn’t be true.)

Find the order for the numbers 1 to 4 where crossing out any single number except the 1 always leaves the remaining sequence in middle-largest-smallest order.

If we write down the numbers 1 to 5 (in any order), you would have to cross out two numbers to leave a sequence of three (these can also be in any order, not necessarily middle-largest-smallest). How many different ways can you do this?

Find an order for the numbers 1 to 5 where more than half of the ways of crossing out two numbers leave the remaining sequence in middle-largest-smallest order. (There are three possible sequences.)

Solution next week

#31 Fold and cut

Solution

For two folds, there will be one hole (in the centre – the other cuts won’t make true holes, just notches on the page edge). For three folds, there will be three holes (one in the centre, plus one above it and one below it along the centre line). As we increase the number of folds to 4, 5, 6 and 7 we get 9, 21, 49 and 105 holes in the page.

If the number of horizontal folds is H and the number of vertical folds is V, you make a grid of creases measuring 2^H – 1 by 2^V – 1 (since each fold doubles the number of creases in that direction, and we subtract one as we don’t count holes on the edge). For example, with 5 folds, we will have folded in half vertically three times and horizontally twice (in the order VHVHV), making 2³ – 1 = 7 lines one way and 2² – 1 = 3 the other, with 3 × 7 = 21 intersections.