Okay -- removing the magic. There are 25 numbers. You choose the color.
Green - 9, 12, 16, 23, 25
Blue - 3, 4, 10, 17, 18
Black - 7, 8, 11, 19, 24
Red - 2, 5, 13, 15, 20
Pink - 1, 6, 14, 21, 22
Nothing else matters until the house you pick. Each of the houses contains one of the numbers from each of the colors, plus one number without any colors assigned.
You tell the program what color your number is, then you tell it what house it's in, isolating the exact number, which is behind the first door selected.