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#33 Number triangles

set by Peter Rowlett

There are two ways (shown below) to arrange the numbers 1 to 3 in a triangular grid so the number at the top of the triangle is the (positive) difference between the two making up the base. (We can mirror each arrangement right-to-left, but won’t count these mirror images as different.)

Can you complete these grids (below), where each triangular grouping of three squares follows the same rule? The first two triangles use the numbers 1 to 6 and the third uses the numbers 1 to 10.

There are four ways to fill a triangle with the numbers 1 to 10. Two have a 3 at the top and two have a 4 at the top. One of these has been started in the example above – can you find the other three?

Solution next week

#32 Crossed out of order

Solution

The sequence 2, 3, 4, 1 is in middle-largest-smallest order if any number other than 1 is crossed out.

There are 10 ways to cross out two numbers from a sequence of five. You have five choices for the first number and four for the second, giving 20 options. But then you must divide by two to remove instances of the same numbers in reverse order.

One order for which more than half of the ways of crossing out work is 3, 4, 5, 2, 1. This is in middle-largest-smallest order after crossing out in these six ways: 3 4 X 2 X; 3 4 X X 1; 3 X 5 2 X; 3 X 5 X 1; X 4 5 2 X; X 4 5 X 1. Other orders that work are 3, 4, 5, 1, 2 and 2, 3, 4, 5, 1.