Spoiler for New Scientist Enigma 1751: “Pentagon of squares” (Follow the link to see the puzzle.)
I’m not happy about my solution to this puzzle. I think I have the right answer, but I had to resort to a computer program to get it. I’m probably thinking about it stupidly.
Here’s a picture of what’s going on:
The red line segments are the pentagon; as required, it doesn’t contain the center of the circle it’s inscribed in . The interior angles (angles between the red line segments) are marked a2, b2, c2, d2, and x. Since the non-square angle x is the smallest, I’ve made it one of the angles on the long side. We know all of the interior angles are less than 180°, and all the ones that are squares must be larger than x and therefore larger than 1, so a, b, c, and d must be in the range [2..13]. Also marked are the radii through each vertex, and the radial angles (between the radii) α, β, ɣ, and δ.
We know the interior angles of a pentagon must sum to 540°, but that’s not enough of a constraint; we also have to make sure the pentagon can be inscribed in a circle. For that we use a theorem that says interior angles subtending the same chord are equal to one another, and to half the central angle subtending that chord. So for instance interior angle a2 and central angle α + β + ɣ subtend the same chord, so a2 = (α + β + ɣ)/2. Likewise x = (β + ɣ + δ)/2, and similarly (pun intended) b2 = 180 – (ɣ + δ)/2, c2 = 180 – (β + ɣ)/2, and d2 = 180 – (α + β)/2. From these relations we can get expressions for the radial angles:
α = 2(a2+c2)–360
β = 720–2(a2+c2+d2)
ɣ = 2(a2+d2)–360
δ = 720–2(a2+b2+d2)
So given choices for a, b, c, and d, we can calculate α, β, ɣ, δ, and x. We require all of these to be positive; x < a2, b2, c2, and d2; and α+β+ɣ+δ < 180 to make the center lie outside the pentagon.
There may be a clever way from here to find the one and only solution, but I don’t see it. Instead I wrote a Perl script that came up with:
a2 = 64, b2 = 169, c2 = 144, and d2 = 121; then α = 56, β = 62, ɣ = 10, and δ = 12 (summing to 140), and x = 42. That in fact is exactly the pentagon I’ve illustrated above. In case you’re suspicious, the interior angles do indeed sum to 540.The answer they’re looking for is: 42, 64, and 121.