This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.
Coming and Going
Add up the number of doors leaving each room; call the sum \(S\). As the number in each room is even, \(S\) will be even. Each interior door has been counted twice (as they can be seen in two rooms) and each exterior door has been counted once. Let \(I\) be the number of interior doors and \(E\) be the number of exterior doors. We have:$$S=2I+E$$$$E=S-2I$$
\(S\) and \(2I\) are even, so \(E\) must be even.
If the number of doors in each room is odd, is the number of exterior doors odd or even?
Let \(S\) be the area of the large square, \(T\) be the area of one of the large triangles, \(U\) be one of the red overlaps and \(V\) be the uncovered blue square. We can write:$$S=4T-4U+V$$
as the area of the square is the total of the four triangles, take away the overlaps as they have been double counted, add the blue square as it has been missed.
We know that \(4U=V\), so:$$S=4T-V+V$$$$S=4T$$
Therefore one of the triangles covers one quarter of the square.
Five congruent triangles are drawn in a regular pentagon. The total area which the triangles overlap (red) is equal to the area they don't cover (blue). What proportion of the area of the large pentagon does each triangle take up?
\(n\) congruent triangles are drawn in a regular \(n\) sided polygon. The total area which the triangles overlap is equal to the area they don't cover. What proportion of the area of the large \(n\) sided polygon does each triangle take up?