Martin Gardner (1914 - 2010) published his 12 matchstick puzzle in the November 1957 Issue of Scientific American (SciAm) The puzzle was republished by SciAm on 2nd May 2026. The puzzle (slightly rephrased by me) states:
Slide 1:
The puzzle has been very popular, but it's solution has a twist that can be quite frustrating until you find it. The solution is available in Appendix 1 (If you want to attempt it before seeing the solution then try to start by first forming a 3,4,5 Pythagorean triangle of area 6 square units).
The 12 matchstick puzzle raises several questions - for example,
(a) What is the maximum and minimum area of polygons that may be formed with the 12 matchsticks?
(b) Are there other solutions to the problem with polygon area equal to 4 sq units?
(c) Can one form polygons of area 3 sq units?
(d) Can one form polygons of area 2 sq units?
etc.
I shall address these questions in the following but also see reference.
Maximum Area: We are in good territory here - as all 12 matchsticks have the same length, the 12-sided polygon formed is a regular polygon. The iso-perimetric theorem states that for n-sided polygons (n-gons) with the same perimeter, the regular polygon (all sides of the same length) has the largest area.
The 12 sided polygon formed is called a dodecagon or 12-gon. It follows from the reference that a regular 12-gon approximating a circle will have the maximum enclosed area. This is calculated in the next slide:
Slide 3:
Minimum Area: There is very little guidance available here - except by trying different configurations. The slide shows an example that, in principle, gives an area approaching zero if matchsticks have zero width.
Slide 4:
In the following, we shall expand the scope of the puzzle and find solutions for enclosed area to be an integral number from 2 to 9 sq units. Generally, there is more than one way to achieve a particular value of the enclosed area. The idea I have followed is to start with a simple geometric shape and explore the way to modify the shape to achieve the desired enclosed area.
1. Rectangular Shapes: There are three possible rectangles that one may form with 12 matchsticks: of sides (5,1), (4,2) and (3,3). We look at them individually:
Rectangle with sides 5 and 1 units: This is a simple rectangle with enclosed area equal to 5 sq units. This is shown in the figure below:
Slide 5:
Rectangle with sides 4 and 2 units: This is a rectangle with enclosed area equal to 8 sq units. However, by moving sticks as shown in slides 6 and 7, it is possible to obtain enclosed area of 8, 7, 6 and 5 sq units.
Slide 5: 
Rectangle with sides 3: This is a square with enclosed area equal to 9 sq units. However, by moving sticks as shown in slide 7, it is possible to obtain enclosed area of 8, 7, 6 and 5 sq units. By using equilateral triangles as in slide 5, we can also obtain a 12-gon of enclosed area equal to 4 sq units.
Slide 7:
2. Triangular Shapes: The 12-gon may be arranged as a Pythagorean triangle with side lengths of 3, 4 and 5 units. The area enclosed in the right-angled triangle is 6 sq units. By moving sticks, It is possible to construct 12-gons of area 5, 4, 3 and 2 sq units. The cases for 5, 3 and 2 sq units are shown in slides 8 to 11.
Slide 8:
I have only given examples of some simple structures, but it possible to construct more complicated looking designs - especially with the use of equilateral triangles to add and subtract areas to a basic polygon. I have desisted from adding several slides that are quite complex. I am, however, not able to construct 12-gons with enclosed areas of 10 and 11 square units. I believe that one will have to start with a dodecagon shape and work from there. Any suggestions will be very welcome.
Thanks for reading - please pass on the link of this blog to others who you feel might enjoy the mental exercise!
Appendix 1
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