Rose and Ivy are logic-savvy landscapers tasked with creating a rectangular garden. Their client has told Rose the desired width and Ivy the desired length (both positive integers). One day their client leaves a voicemail stipulating that the scale of this project must be exactly 12. In this landscaping company, the “scale” can refer to either the area or the perimeter of the garden, but the client doesn’t specify which. The following conversation ensues:
Rose: “I know the desired width, but even hearing a scale of 12, I still don’t know the length.”
Ivy: “I know the desired length, but even hearing you say that, I don’t know the width.”
Rose: “Alas, I’m still stumped.”
What is the client’s desired width for the garden?
The width is 3. Call the width W and the length L. Both landscapers know that either W × L = 12 or 2 × (W + L) = 12; that is, either the area or the perimeter of the garden equals 12.
Consider Rose’s perspective. Knowing the width and the scale is not enough for her to deduce the length. If she knew the width and the scale and whether the scale referred to area or perimeter, then she could always deduce the length by solving the relevant equation for L. So knowing all she knows, it must still be ambiguous to her whether the scale refers to area or perimeter. In particular, this means that W must be a factor of 12. If it weren’t, that would rule out area because factors of 12 are the only integers that can multiply to 12. Furthermore, W must be less than 6 because otherwise the perimeter would exceed 12. This narrows the possible widths to {1, 2, 3, 4}, and now, after Rose’s statement, Ivy can deduce this as well.
Ivy still can’t figure out the width, which means by the same argument, we can narrow her possible length values down to {1, 2, 3, 4}. The length can’t be 1, because none of Rose’s remaining possible widths could yield an area or perimeter of 12 with a length of 1. The length also can’t be 2: if it were, Ivy would know the width was 4, creating a perimeter of 12. (A length of 2 can’t form an area of 12 with any of Rose’s remaining widths.) So the length can only be 3 or 4.
After Ivy’s response, Rose now knows that the only possible lengths are 3 and 4, yet she still cannot deduce which one is correct. This means the width must be able to form an area of 12 and a perimeter of 12 with the remaining possible lengths. W = 2 and W = 3 are the only remaining ways to form a perimeter of 12 (L would equal 4 and 3, respectively), whereas W = 3 and W = 4 are the only remaining ways to form an area of 12. (L would equal 4 and 3, respectively.) W = 3 belongs to both cases, so it is the only value for which Rose would still be unsure.
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