- A box, having the precise shape of a rectangular cuboid, has integer dimensions.
- When each one of the dimensions is increased by 3, the volume of the box trebles.
Determine the largest possible dimension of this box.
The smallest dimension cannot be 1, since adding 3 would more than quadruple the box size.
The smallest dimension must be less than 7: Three times the volume of a7x7x7 box is already larger than a 10x10x10 box and increasing the dimensions just makes the volume difference larger.
So the smallest dimension is 2, 3, 4, 5, or 6.
Let the box have dimensions x, y, z. The equation implied in the problem then is 3xyz = (x+3)(y+3)(z+3).
Let x be constrained to 2, 3, 4, 5, or 6. Then the problem can be split into five cases:
x=2: 6xyz = 5(y+3)(z+3)
x=3: 9xyz = 6(y+3)(z+3)
x=4: 12xyz = 7(y+3)(z+3)
x=5: 15xyz = 8(y+3)(z+3)
x=6: 18xyz = 9(y+3)(z+3)
x=2: (y-15)*(z-15) = 270
x=3: (y-6)*(z-6) = 54
x=4: (5y-21)*(5z-21) = 756
x=5: (7y-24)*(7z-24) = 1080
x=6: (y-3)*(z-3) = 18
Since we are interested in the largest possible dimension, we only need to consider the widest factorization that lets us solve in integers:
x=2: y-15=1 and z-15 = 270 -> y=16 and z=285
x=3: y-6=1 and z-6=54 -> y=7 and z=60
x=4: 5y-21=4 and 5z-21=189 -> y=5 and z=42
x=5: 7y-24=4 and 7z-24=270 -> y=4 and z=42
x=6: y-3=1 and z-3=18 -> y=4 and z=21
The largest of these is our answer 285.
Analytic Solution
