Problem number 13 states. The numbers from 1 to 6 are placed in the circles at the intersections of three rings. The position of number 6 is shown. The sum of the numbers on each ring are the same. What number is placed in a circle with a question mark? So first let's name all the circles around with variables. We'll call them W, X, Y, and Z. First let's consider this ring in green. Let's call the sum of all of the circles on this ring A. So W plus X plus Y plus Z. We do this because the problem tells us the sums of the circles on each ring are the same. So these numbers must be important. Next let's do a sum for this next ring. We'll call this sum B, but it's important to note that the value of this sum is identical to that of the first ring. We know this because the problem tells us this. Next let's take a look at the third ring. We'll call this sum C. This one's going to be equal to X plus Z plus the number we're figuring out plus 6. At this point this problem gets a little bit tricky and the next move might not really be obvious. In our case what we're going to do is we're going to sum up all the circle's values and call that sum S. From here we will do some substitution. So here we're going to grab three copies of the S equation and each one we're going to replace a different part of S with A, B, or C. So in the end we get S equals A plus question mark plus 6, S equals B plus X plus Z, S equals C plus W plus Y. But now what we must remember is that A equals B which equals C. This means that the sum of the pairs of numbers not in the common ring must be equal to one another. So this is how that would look. The sum of the red circles is equal to that of yellow and to that of green. Let's figure out the sum value by adding up all the possible numbers that can be in circles and then dividing by the number of pairs. So let's look at the number that we must put in the circles which are 1 through 6. Look at their sum which is going to be 21. Divide by 3 since we have 3 pairs and we get a sum for each of 7. This means the mystery number must be 1 because 1 plus 6 is going to be equal to 7. So the question asked us, what number is placed in the circle with the question mark? The answer is 1, letter A.

number arrangement

ring intersections

equal sums

puzzle solution

number pairs