The clockmaker's birthday cake
Problem
The clockmaker's wife was a mathematician. On his birthday she invited five other couples to tea. She made a birthday cake decorated to look like a clock face with numbers made from pink icing.
She cut up the cake into twelve slices with a number on each slice:
The slices that she gave each couple added to the same number. What was the number?
The clockmaker had the slice with $12$ on it because it was his birthday. One guest wanted $7$ because it was her lucky number. All the slices that went to men added to a number equal to those that went to the women.
How could this be arranged?
Getting Started
Student Solutions
Charles sent in his solution:
All the numbers together add up to $1+2+3+...+12=78$. There are six couples, each with the same total amount, so this must be $78/6=13$. So each couple's numbers add up to $13$.
I'm going to write the pairs in brackets, like this ( , ). The first number is the one for the man, and the second for the woman. So we know we have $(12, 1)$ and $(6, 7)$. We want the ladies' numbers and the men's numbers each to add up to $78/2=39$. Here's a way we could do it:
$(12, 1)
(6, 7)
(2, 11)
(10, 3)
(5, 8)
(4, 9)$
And $12+6+2+10+5+4=39=1+7+11+3+8+9$.
Teachers' Resources
The Clockmaker's Birthday Cake
The clockmaker's wife was a mathematician. On his birthday she invited five other couples to tea. She made a birthday cake decorated to look like a clock face with numbers made from pink icing.
She cut up the cake into twelve slices with a number on each slice:
The slices that she gave each couple added to the same number. What was the number?
The clockmaker had the slice with $12$ on it because it was his birthday. One guest wanted $7$ because it was her lucky number. All the slices that went to men added to a number equal to those that went to the women.
How could this be arranged?