Another week, another riddler. This was a fun one from Steve Abney:

*Suppose I have a rectangle whose side lengths are each a whole number, and whose area (in square units) is the same as its perimeter (in units of length). What are the possible dimensions for this rectangle?*

*Alas, that’s not the riddle — that’s just the appetizer. The rectangle could be 4 by 4 or 3 by 6. You can check both of these: 4 · 4 = 16 and 4 + 4 + 4 + 4 = 16, while 3 · 6 = 18 and 3 + 6 + 3 + 6 = 18. These are the only two whole number dimensions the rectangle could have. (One way to see this is to call the rectangle’s length a and its width b. You’re looking for whole number solutions to the equation ab = 2a + 2b.)*

*On to the main course! Instead of rectangles, let’s give rectangular prisms a try. What whole number dimensions can rectangular prisms have so that their volume (in cubic units) is the same as their surface area (in square units)?*

*To get you started, Steve notes that 6 by 6 by 6 is one such solution. How many others can you find?*

To solve this I set the volume equation

To be equal to the surface area equation

Rearranging, we can solve for **a**

Then we iterate through the the first ten thousand integers for **b** and **c** to see if it leads to an integer value for **a**. It works for the following values:

(3, 8, 24)
(3, 9, 18)
(4, 8, 8)
(4, 6, 12)
(6, 6, 6)
(3, 12, 12)
(3, 10, 15)
(3, 7, 42)
(4, 5, 20)
(5, 5, 10)

Do I conclusively know that these are the only ones? No. But the fact that there aren’t any between 50 – 10,000 means I don’t think there are any after 10,000… but I’d be excited to be proven wrong!

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