Problem number 18 states, a 3x3 square initially has the number 0 in each of its cells. In one step, all four numbers in one 2x2 sub-square, such as the shaded one for example, are then increased by 1. This operation is repeated several times to obtain the arrangement on the right. Unfortunately, some numbers in this arrangement are hidden. What number is in the square with the question mark? Here we have how the square started out initially. Let's simulate the step where all the top right squares numbers increase by 1. We'll say this event happens A times. That means each of those squares would have a value of A. Then let's say that this sub-square is touched B times. So we would add B to each square that is inside of this sub-square. Let's do the same for this square, saying that it is stepped C times. So we would again add C to each of the squares inside of it. Now let's do the same for this one. We'll say this one is pressed D times, and add D to each of the squares. So now let's transform these expressions into equations for the squares that we know the values of. So for the top square, we would have 18 equals A plus B. For the middle one, we would have 47 equals A plus B plus C plus D. For the bottom left one, we would have D equals 13. And the bottom right one, well that's the mystery number that we're trying to figure out. So now we have four equations, so let's try and solve for C. First we can notice that we can substitute D equals 13 into the middle square equation formula. We can also substitute A plus B equals 18 into the middle square formula. If we do this, we get 18 plus C plus 13 equals 47. Subtract 18 and 13 from both sides, we get C equals 16, which is our mystery number. So the question asked us, what number is in the square with the question mark? The answer is 16, letter C.

3x3 square

2x2 sub-square

system of equations

operation frequency

solution