Another Nine-Dots Puzzle
I recently wrote an essay, Thinking Inside and Outside the Box, which starts with a famous nine-dots puzzle that kicked off the expression: thinking outside the box. Here is another puzzle with the same nine-dots setup.
Puzzle. What is the smallest number of squares needed to ensure that each dot is in its own region?
![9 dots puzzle](http://tanyakhovanova.com/BlogStuff/ThinkingOutsideTheBox/9dots2Sm.png)
Usually, people who try to solve this puzzle come up with the following four-squares solution.
![9 dots puzzle non-solution](http://tanyakhovanova.com/BlogStuff/ThinkingOutsideTheBox/9dots2Solution1Sm.png)
As with the classic nine-dots puzzle, they imagine that the dots are on a grid and try to build squares with sides parallel to the grid lines. What would be the outside-the-box idea? The sides of the squares would not need to be parallel to the grid. This way, we can solve the puzzle with three squares.
![9 dots puzzle solution](http://tanyakhovanova.com/BlogStuff/ThinkingOutsideTheBox/9dots2Solution2Sm.png)
One of my MathRoots students offered a different and awesome solution also using three squares.
![9 dots puzzle another solution](http://tanyakhovanova.com/BlogStuff/ThinkingOutsideTheBox/9dots2Solution3Sm.png)
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JBL:
Is it clear that 2 is impossible? (The most naive thing doesn’t work: you get 10 regions if you take two squares of the same size and same center that are rotated 45 degrees.)
[Sorry if this comes through multiple times.]
5 August 2022, 9:08 pmAlex:
is this a strictly 2D problem, with squares always in the same plane as the dots?
5 August 2022, 10:58 pmif squares are allowed to be “square” on some other planes and projected on this one, other solutions might be possible
Ivan:
By “thinking outside the box” those two solutions might be generalized:
7 August 2022, 3:27 amSolution 1: The edges of the red square might intersect the edges of the blue one.
Solution 2: Three edges of the blue square might be intersected by the edges of the red and green squares.
rosie:
It doesn’t surprise me that some people come up with your first solution attempt. Your statement of the problem did not make it clear what exactly is or is not allowed as a “square”, nor yet what you mean by “region”.
Thinking outside the box is all very well, but the problem statement should be clear and unambiguous in every aspect.
8 August 2022, 8:30 amMath Book:
I’m not surprised that some folks come up with your initial effort as a solution. You did not explicitly define what is or is not permitted as a “square” or what you mean by “region” in your presentation of the issue.
14 August 2022, 1:13 amJBL:
This comment is just to point out that the previous comment is an old-style spam-bot
31 August 2022, 10:14 am