Edited by Richard Walker, Monday, 9 June 2025, 00:29
Here is the solution to "The Two Squares", which asked given squares of side 3 and 4 arranged as shown, what is the are of the overlap? The answer is 4 square units. Here are two ways to see this.
1. The question doesn't give any information about the orientation of the small square, yet we are asked to find the area in question. This suggests that orientation makes no difference (it's one of those questions where absence of a piece of information is itself a clue). So we can arrange the square however we please and choosing the orientation shown below below tells us the overlap is one quarter of the larger square.
2. Alternatively, let's add three copies of the smaller square and now the fact that the overlap is one quarter of the small square, irrespective of the orientation, becomes obvious.
In fact what makes it true that the overlap is one quarter of the smaller shape is not that is a square, it is that it has 4-fold rotational symmetry. The square can be replaced with any arbitrary shape so long as it has this property. Here's a pinwheel, for example:
Solution to "The Two Squares"
The answer is 4 square units. Here are two ways to see this.
1. The question doesn't give any information about the orientation of the small square, yet we are asked to find the area in question. This suggests that orientation makes no difference (it's one of those questions where absence of a piece of information is itself a clue). So we can arrange the square however we please and choosing the orientation shown below below tells us the overlap is one quarter of the larger square.
2. Alternatively, let's add three copies of the smaller square and now the fact that the overlap is one quarter of the small square, irrespective of the orientation, becomes obvious.
In fact what makes it true that the overlap is one quarter of the smaller shape is not that is a square, it is that it has 4-fold rotational symmetry. The square can be replaced with any arbitrary shape so long as it has this property. Here's a pinwheel, for example: