Question 18. Annie wants to place the numbers 1 to 10 in the circles in the diagram, with one number in each circle. She wants the sum of the numbers in any four circles that are in a straight line, for example the four grey circles, to be 23. What number must she place in the circle with a question mark? We have three straight lines, which we are told each must sum to 23. That means the sum of all three lines together must be 3 times 23, which is 69. Notice how when we do this, we add up all the circles, but we also end up counting the number in the question mark circle three times, as all three lines share this circle. This means that 69 is the sum of all the circles, plus two extra copies of the question mark circle. We also know that the numbers 1 through 10 are placed in each of the circles, which means that the sum of all the circles is 55. Notice how if we subtract these two values from each other, we are left with the value of the two copies of the question mark circle. We can divide this value, 14, by 2 to get that the number that goes in the question mark circle to be 7.

circle puzzle

number placement

sum to 23

math problem

solution 7