Question number 14, Paul entered positive integers into a 3x3 grid in such a way that the sum of the numbers in each 2x2 square is equal to 10. Four of the numbers are shown in the illustration. Which of the numbers below represents a possible sum of the remaining five numbers? So here we have a grid with the numbers filled in, and let's give names to the remaining empty spots. So let this be A, B, C, D, and E. And then let's sum four squares at a time. So in the upper left we have A plus C plus 3, and that's equal to the upper right, B plus C plus 5, that's equal to the lower left square, C plus D plus 5, and one more, C plus E plus 7, all of them equal to 10. So somehow we have to obtain the sum of these numbers, A plus B plus C plus D plus E, and The way we will do that is to add up all of these four equations. So this is what we have to do. We have four of these equations, so each of them equal to 10, the sum of the four is 40, and we have A plus C plus 3 from the first square, then B plus C plus 5, C plus D plus 5, and C plus E plus 7. So on the right hand side here, we have one A, and then four C's, one B, one D, and one E. And the numbers add up to 3 plus 5 plus 5 plus 7, that gives us 20. And so subtracting the 20, we obtain 20 is A plus B plus 4C plus D plus E, which is not exactly what we want, so we will subtract 3C from both sides, 20 minus 3C is A plus B plus C plus D plus E, which is the quantity we're trying to study here. So we can fill that in, 20 minus 3C, and that has to be equal to one of the numbers given in the choices. Let's factor out the 3, and we will have A6 minus C plus remainder 2. And that's the important thing to notice, that one of the numbers here in the choices upon division by 3 has to produce a remainder of 2. So let's go through the choices and see what's going on. 9 is equal to 3 times 3 plus remainder 0, 10 is 3 times 3 plus remainder 1, 12 is divisible by 3, we have 3 times 4 plus 0, 3 times 4 plus 1 gives us 13, and none of the choices produce a remainder of 2. So we conclude that none of the numbers listed here represent a possible sum of the remaining five numbers, and so the answer must be E, none of these.

3x3 grid

2x2 square

sum formula

integer puzzle

remainder condition