A riddle about cake

One of the patterns of life at my firm is the birthday email: cakes or chocolate brought in to celebrate an event such as a birthday, or return from holiday, etc.

Most of such emails are routine: and sent to the people in the department or grouping. Occasionally, they are sent to 'UK firm all' which would be a morsel per person; even more rarely the emails themselves are interesting.

One such was received last September from a mathematically minded colleague.

On Sunday I advanced another year more

To the sum of all quarters from nought til 4

Come over and enjoy a nice slice of cake

(though I admit it is one that I did not make)

I sent my colleague a happy birthday note, and also asked whether my calculation of her age was correct (amongst mathematicians, this can't be rude?). Disappointingly, I was the only person who gave her the correct answer: not good for a firm of accountants?!

How old is she?

Answer

The riddle is asking us to sum 0/4, 1/4, 2/4….up to 16/4. Most colleagues thought this has to be done long windedly (moan warning: why do people <30 use calculators for every calculation?). But the sum is 1/4(1+2+3+…+16) and the part in brackets is 16*17/2 by the formula attributed to Gauss, n*(n+1)/2.

So, my colleague is 1/4*16*17/2, or 2*17, 34.

Gauss's formula can be derived in a number of ways. The one I prefer is to list the numbers 1 to (say) 16 out in a line: 1, 2, 3….16. Then write them in reverse in the line below. 16, 15,14…1. Then add the two rows up: 17, 17, 17…17. So, twice the sum is sixteen lots of seventeen, so the sum is n(n+1)/2.

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