Shogi shapes
Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring?
Age
11 to 14
Challenge level
Problem
Can you work out the angles in the pictures below?
On the left, a regular octagon is surrounded by congruent kites. On the right, a regular decagon is divided into two different types of rhombuses.
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Here are two more polygon puzzles to try.
The first one is a regular nonagon drawn inside an equilateral triangle. The second is also a regular nonagon, with lines joining vertices to each other and to the centre.
Can you work out the angles?
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The game of Shogi, also known as Japanese Chess, is played with tiles on a 9 by 9 board.
Here is a picture of some Shogi pieces that have been arranged in a pattern:
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What questions might a mathematician ask?
Here are some that we thought of:
- Is there a quick way to count the number of tiles in the picture?
- What shape are the tiles?
- Which angles can be calculated?
- If we assume that the tiles are symmetrical, there are two pairs of matching angles and one other angle. What is the range of possible sizes of that "lonely" angle?
- Could we make other sets where we need more tiles to go all the way round with no gaps?
Can you come up with answers to any of your questions?
Can you justify your answers?
Thanks to Percy Henderson who shared with us his exploration of Shogi tiles and kindly granted permission to use his picture, and to Don Steward for the ideas behind the other polygon puzzles.
Getting Started
Here are a couple of methods you could use to work out the interior angles of a regular nine-sided polygon (a nonagon):
1.
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Imagine taking a walk around a regular nine-sided park.
The second picture shows the angle you need to turn through when you reach the first corner.
The third picture shows all the angles you turn through when you walk once round the park.
What angle do you turn through altogether?
What must each exterior angle (the red angles) be?
What must each interior angle be?
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2. Here is a regular nonagon:
It has been divided into nine isosceles triangles.
What is the total of all the angles in all the triangles?
Now subtract the angles at the centre of the nonagon to work out the total of all the interior angles of the nonagon.
Can you adapt these methods for any regular polygon?
Student Solutions
Arnav from Franconian International School in Germany began by thinking about angles in a nonagon from the Getting Started section:
What angle do you turn through altogether with a nonagon?
Answer: 360 because 360 degrees is a complete circular turn and a nonagon lands on the same dot you started with every time you turn 360 degrees.
What must each exterior angle (the red angles) be?
Answer: 40 because 360$\div$9 = 40
What must each interior angle be?
Answer: 140 because 180$-$40 = 140
Luxsija from Mill Hill County High School, Year 7 students from Chesterfield High School, Oliver from The Priory Academy LSST and Rishika from Nonsuch High School for Girls, all in the UK, and Ashlynn from ISF Academy in Hong Kong found the angles in the octagon and kites. This is Oliver's work:
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For the decagon and the rhombuses, Rishika and the Chesterfield year 7 students started with the decagon. This is Luxsija's work:
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Ashlynn and the Chesterfield year 7s started with the rhombuses. This is the Chesterfield year 7s' work:
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They used the facts that the angles at a point (in the centre) add up to 360$^\circ$, opposite angles in a rhombus are equal, and angles in a rhombus add up to 360$^\circ$.
For the triangle and the nonagon, Rishika, Luxsija, Ashlynn and Chesterfield year 7 all began by finding the interior angles of each shape. This is Rishika's work:
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As it is an equilateral triangle, angles are equal. Angles in a triangle add to $180^\circ$, and therefore, angle $\text{a}=180\div3 = 60^\circ.$
Sum of interior angles in a nonagon is:
$(n-2)\times180 = (9-2)\times180 = 1260$
Therefore, angle $\text{b}$ is $1260\div9 = 140^\circ$
Angle $\text{c}$ and angle $\text{d}$, which are equal, are $180^\circ-\text{angle b}$ (because they are on a straight line), so $\text{angle b}= 180-140 = 40^\circ$
As angles in a quadrilateral add to $360^\circ$, $\text{angle f}=360 - \text c-\text d-\text a = 360 - 40-40-60 = 220^\circ$
This is verified as $\text f+\text b$ should equal $360^\circ$ as it is around a point ($140+220 = 360,$ so it is correct).
For the other nonagon, Luxsija and Chesterfield year 7s used kites and parallelograms inside the shapes. This is Luxsija's work. The pink shapes are not actually parallelograms, but they are quadrilaterals so the angles still add up to 360$^\circ$:
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Rishika used symmetry and properties of angles in triangles and on lines and points:
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Angle $\text{a}$ is established to be $140^\circ$ (see above).
Angle $\text{b}$ is half of angle $\text{a}$, which is $0.5\times140 = 70^\circ$
Angle $\text{c}$ is $360\div3$ (as 3 fit around a point) $= 120^\circ$
Angle $\text{d}$ is an angle in an equilateral triangle, and so must be $60^\circ$ as all angles are equal and angles in a triangle add to $180^\circ$
Angle $\text{e}$ is equal as it is vertically opposite, and therefore is $60^\circ$ as well.
Angle $\text{f}$ is part of an equilateral triangle $= 60^\circ$.
Angle $\text{g} =\text a - \text f$ as $\text{g} = 140 - 60 = 80^\circ$
Angle $\text{h}$ must be $360 - \text c - \text b - \text g$ as angles in a quadrilateral add to
$360^\circ$.
$360 -70-120 -80 = 90^\circ$
Also, angles $\text{g , d}$ and $\text{a}$ can be verified as they form a kite and must add to $360^\circ$. $80+80+140+60
= 360^\circ$ and therefore, they are correct.
Angle $\text{i}$ must be $(360 - 60-60)\div2$ as angles around a point add to $360^\circ$.
Angle $\text{i} = 120^\circ$.
Angle $\text{j}$ must be $360^\circ - 70 - 80 - 120 = 90^\circ$
Ashlynn used a pentagon:
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The interior angle of nonagon is $180-40=140^\circ$
In the red pentagon, there are three $140^\circ$ angles. The small angle is equal to $(540^\circ-3\times140^\circ)\div2=60^\circ$
All the small triangles have angles of $60^\circ$ and are equilateral.
Luxsija, Rishika and Ashlynn all counted the shogi tiles and found some angles. This is Rishika's work:
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The tiles are in pairs - there are $9$ pairs. $2\times9 = 18.$
The tiles are congruent, irregular pentagons.
The shape inside is $36$-sided (each tile provides $2$ sides), so the sum of the interior angles is:
$(36-2)\times180^\circ=6120^\circ$
However, we have to remember that this includes $18$ angles [smaller than $90^\circ$] and $18$ [reflex] angles.
We can work out angle $\text a$ because the shogi pieces are arranged to make a circle - $360^\circ$. There are $18$ tiles, so $18\text a=360^\circ\Rightarrow\text a = 20^\circ$
As the triangle formed is isosceles, we can work out angle $\text b$ as angles in a triangle add to $180^\circ.$
$\text b=(180-20)\div2=80^\circ$
$3\times\text{angle c}$ fits around a point, and therefore it must be $120^\circ.$
From there, Rishika found angle $\text d$ from the sum of the interior angles in a pentagon, and angle $\text e$ from the sum of angles at a point. However, angle $\text c$ is not neccessarily $120^\circ.$ This is Ashlynn's work on the angles in the tiles:
If we assume that the tiles are symmetrical, there are two pairs of matching angles and one other angle. What is the range of possible sizes of that "lonely angle" (angle $\text c$)?
The smallest angle is when it becomes a triangle i.e. $20^\circ$ (so angle $\text c$ is at angle $\text a$)
The biggest angle is when it becomes a trapezoid i.e. $180^\circ$
Teachers' Resources
Why do this problem?
This problem offers an enticing context within which to work on angles and regular polygons. Unlike an exercise in which students may be invited to calculate particular angles for no apparent reason, the images in this problem are, we hope, intriguing enough to provoke students' curiosity so that they ask "I wonder what angles are needed to create this shape? I wonder if I could recreate it for myself?"Possible approach
Begin with the first four shapes and challenge students to work out as many different angles as they can. You may wish to print out this worksheet: Shogi Shapes.As students are working, listen out for interesting and useful insights that they are using to work out the angles, and share them in a short plenary.
Then show this image:
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Invite students to come up with questions to explore based on the image. Some suggested questions are listed below. Then choose as a class which questions to work on. Finally, bring the class together to discuss what they have found out about this image.
Key questions
Is there a quick way to count the number of tiles in the picture?What shape are the tiles?
Which angles can be calculated?
If we assume that the tiles are symmetrical, there are two pairs of matching angles and one other angle. What is the range of possible sizes of that "lonely" angle?
Could we make other sets where we need more tiles to go all the way round with no gaps?
Possible extension
Students could use ruler, compasses and protractors, or dynamic geometry software such as GeoGebra, to recreate the images from the problem.Possible support
The Angles, Polygons and Geometrical Proof short problems collection might be a useful source of questions to prepare students for this task.The advice offered in Getting Started might also be helpful.