Question 21. Jenny decided to enter numbers into the cells of a 3x3 table so that the sum of the numbers in all four possible 2x2 squares will be the same. The numbers in three of the corner cells have already been written, as shown. Which number should she write in the fourth corner cell? Let's take a look at our 3x3 grid. We have the numbers 2, 3, and 4. Now we know that any 2x2 is equal to every other 2x2. So let's cover the bottom row, and we notice that we have 2 and 4. But every other cell they share is the same. So if we call this x, we know that this has to be x plus 2, since the other two cells in between are equal to each other. So the difference has to be made up in the remaining cell. Next, let's remove the leftmost column and do the same thing. We see that there is a difference of 1. So let's call this y and y plus 1. And finally, we can label the number in the center as z. Now we can get to work on figuring out the corner cell. When we have this, we know that the total will be y plus x plus z plus 4, equals to the sum of all four cells. And this should be equal to the sum of every other 2x2 square. If we look at this square with the question mark, we get this equation. x plus 2 plus y plus 1 plus z plus our unknown equals the same sum. We can simplify this a little bit, and then we can set it equal to our previous equation. And we get x plus y plus z plus 3 plus our unknown equals x plus y plus z plus 4. We can subtract x, y, and z from both sides, and we get 3 plus our corner piece equal to 4. And with this, we can get our answer by simply doing 4 minus 3, and that gives us the answer of b, 1.

3x3 grid

2x2 squares

algebra

equal sum

unknown cell