That’s Maths: How many gifts did my true love give to me overall?

We can easily count them, but this is tedious and there is a good chance of an error. Here’s a better way

We all know the festive carol The Twelve Days of Christmas. Each day, "my true love" receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, 10 on the fourth, 15 on the fifth, and so on. Here is a Christmas puzzle: what is the total number of gifts over the 12 days?

While you work on that, here is a riddle: why do mathematicians confuse Halloween and Christmas? Because 31 October equals 25 December. If this makes no sense, recall that three times eight plus one makes 25, so the octal number 31 equals the decimal number 25.

Back to the gifts: we can easily count them, but this is tedious and there is a good chance of an error. So let us reason mathematically. The number of gifts on day N is 1 + 2 + … + N, where the dots indicate that we add all numbers from 1 to N. We can check the first few values of N, and we get 1, 3, 6 and 10. In fact, there is a formula for the number of gifts on day N, namely N ( N + 1 ) / 2. Again, we can easily check that this works for the first few values of N.

The numbers of gifts each day are called triangular numbers: they can each be arranged in a triangle. For N = 1 and N = 3 this is obvious. For N = 4, think of a triangle of 10 bowling pins and for N = 5 imagine 15 snooker balls arranged in a triangle.