Question

Draw a clock face with the numbers 1 to 12 in their usual positions. Can you draw a line so that the numbers on one side of it add up to the same as the numbers on the other side?

Answer 1

The sum of all the numbers on the clock face is 78, so you want to find six consecutive numbers that add up to 78÷2=39. If we call the first number n, then the sum of six consecutive numbers is n+(n+1)+(n+2)+(n+3)+(n+4)+(n+5)=39. By now gathering terms we get the equation 6n+15=39 So 6n=24 and n=4 Then the six consecutive numbers are 4,5,6,7,8,9 and so the line runs from after 9 to after 3. As there was only one solution to the algebraic equation, then this is the only solution to the problem.