12 days of Christmas


I had been wondering (as one does at this time of year!), how many presents in total are in the song “12 days of Christmas”.

At first glance, it would appear to be 1+2+3+…+11+12 = 78; but that does not account for all the repeated additional gifts on subsequent days.

So in actual fact, it is 1 + (2+1) + (3+2+1) + … + (12+11+10+…2+1), which can be rearranged as 12×1 + 11×2 + 10×3 + 9×4 + 8×5 + 7×6 + 6×7 + 5×8 + 4×9 + 3×10 + 2×11 + 1×12. Noting the symmetry of 12×1 + 1×12 etc. this can be further simplified to 2x(12+22+30+36+40+42) = 364. One for every day of the year except Christmas, as the eminent mathematician, Marcus du Sautoy noted in his slightly different method of summing the presents.

Of course, on a much lighter note, there is always Frank Kelly’s Christmas Countdown — what it might be like to receive the aforementioned gifts!