Creating 6-15 to 6-19

pages/11-11-hanging-a-picture.md

@@ -26,7 +26,7 @@ Did you know that throughout history, and even to this day, some numbers are ban
 @[youtube](VmWVXOIQblM?end=266&rel=0) _(If you have problems watching the video, right click to open it in a new window)_
 
 ::: problem id=1_1 marks=2
-__1.1__ According to the video, why have numbers been banned historically?
+__1.1__ According to the video, what reasons have been given throughout history for banning numbers?
 
 * [ ] Because numbers can encode just about any information.
 * [ ] Because the existence of some numbers was perceived as a threat to the notion of a perfect universe.
@@ -100,7 +100,7 @@ The `XX^(-1)` and `Y^(-1)Y` at each end cancel, leaving `YXY^(-1)YY^(-1)`.
 
 Then the `YY^(-1)` on the right cancels, leaving `YXY^(-1)`.  
 
-The last `Y` and `Y^(-1)` can't cancel because the `X` is in the middle. 
+The last `Y` and `Y^(-1)` can't cancel because the `X` is in the middle.
 :::
 
 

pages/6-15-constant-width.md

@@ -0,0 +1,96 @@
+# Shapes of constant width
+
+<div class="dictionary">
+
+__Noun__: Parallelogram
+__Pronunciation__: /ˌparəˈlɛləɡram/
+
+1. a portmanteau word combining parallel and telegram. A message sent each
+week by the Parallel Project to bright young mathematicians.
+
+</div>
+
+*	Tackle each Parallelogram in one go. Don’t get distracted.
+*	Finish by midnight on Sunday if your whole class is doing parallelograms.
+*	Your score & answer sheet will appear immediately after you hit SUBMIT.
+*	Don’t worry if you score less than 50%, because it means you will learn something new when you __check the solutions__.
+
+
+## 1. Does a wheel have to be circular?
+
+Watch this video where Steve Mould shows an array of shapes with a peculiar property:  
+
+@[youtube](cUCSSJwO3GU) _(If you have problems watching the video, right click to open it in a new window)_
+
+::: problem id=1_1 marks=2
+__1.1__ Which of these shapes is a solid of constant width?  
+
+* [ ] hexagon
+* [x] circle
+* [ ] triangle
+* [ ] square
+:::
+
+::: problem id=1_2 marks=2
+__1.2__ A triangle can be transformed into a Reuleaux triangle by rounding the edges, which set of polygons can this be done for?  
+
+* [ ] All polygons
+* [ ] Only regular polygons
+* [ ] Polygons with an even number of edges
+* [x] Polygons with an odd number of edges
+:::
+
+::: problem id=1_3 marks=2
+__1.3__ Why are coins made with constant width?  
+
+* [ ] So people aren’t hurt by the sharp edges in their pocket
+* [ ] It is cheaper
+* [x] So they can roll in vending machines
+:::
+
+
+## 2. Cutting Shapes
+
+::: problem id=2_1 marks=2
+__2.1__ If a regular hexagon is cut in half, from point to point, it will produce two of which shape:  
+
+* [ ] Pentagon
+* [ ] Triangle
+* [x] Trapezium
+* [ ] Tetrahedron
+:::
+
+::: problem id=2_2 marks=2
+__2.2__ A regular tetrahedron with edges of length 6 cm has each corner cut off to produce the solid shown.  
+
+![](/resources/6-15-constant-width/2-truncated-tetrahedron.png){image align="center"}  
+The triangular faces are all equilateral triangles, but not necessarily the same size.  
+
+What is the total length of the edges of the resulting solid?
+
+<input type="number" solution="36"/> cm
+
+^^^ hint id=2_2_1 marks=1
+Do you notice any lengths which are the same?  
+
+![](/resources/6-15-constant-width/2-truncated-tetrahedron-hint.png){image align="center"}
+^^^
+
+---
+Although the tetrahedron (triangle based pyramid) has been truncated, the total edge length remains the same.  
+
+6 edges at 6cm each = 36cm
+:::
+
+
+Before you hit the SUBMIT button, here are some quick reminders:
+
+*	You will receive your score immediately, and collect your reward points.
+*	You might earn a new badge... if not, then maybe next week.
+*	Make sure you go through the solution sheet – it is massively important.
+*	A score of less than 50% is ok – it means you can learn lots from your mistakes.
+*	The next Parallelogram is next week, at 3pm on Thursday.
+*	Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.
+
+Cheerio,
+Simon & Ayliean.

pages/6-16-wobbly-circles.md

@@ -0,0 +1,114 @@
+# Wobbly circles
+
+<div class="dictionary">
+
+__Noun__: Parallelogram
+__Pronunciation__: /ˌparəˈlɛləɡram/
+
+1. a portmanteau word combining parallel and telegram. A message sent each
+week by the Parallel Project to bright young mathematicians.
+
+</div>
+
+*	Tackle each Parallelogram in one go. Don’t get distracted.
+*	Finish by midnight on Sunday if your whole class is doing parallelograms.
+*	Your score & answer sheet will appear immediately after you hit SUBMIT.
+*	Don’t worry if you score less than 50%, because it means you will learn something new when you __check the solutions__.
+
+
+## 1. Keep on rolling!
+
+Circles, famously, roll pretty well.  
+
+Watch this video where Matt Parker joins two circles to make a wobbly shape, which still rolls surprisingly well.  
+
+@[youtube](OEMA6jhi5Qo) _(If you have problems watching the video, right click to open it in a new window)_
+
+::: problem id=1_1 marks=1
+__1.1__ When rolling a circle, what happens to its center of mass?  
+
+* [x] It stays at a constant height
+* [ ] It moves up and down
+* [ ] It remains in contact with the surface its rolling on
+:::
+
+::: problem id=1_2 marks=2
+__1.2__ When rolling a square, what happens to its center of mass?  
+
+* [ ] It stays at a constant height
+* [x] It moves up and down
+* [ ] It remains in contact with the surface its rolling on
+:::
+
+::: problem id=1_3 marks=2
+__1.3__ In the wobbly circles, where is the center of mass?  
+
+* [ ] It changes from one center to the other
+* [x] The midpoint between the centers
+* [ ] It has two centers of mass
+:::
+
+::: problem id=1_4 marks=2
+![](/resources/6-16-wobbly-circles/1-penny-farthing.png){image align="right"}
+__1.4__ On a penny farthing bicycle the big wheel has a circumference of 3m and the small wheel has a circumference one quarter the size of the big wheel.  
+
+After the bike has traveled 9m, what is the total number of revolutions of both wheels combined?  
+
+* [ ] 9
+* [ ] 4
+* [x] 15
+* [ ] 0
+
+---
+
+Wheels turn along their circumference, so after 9m the big wheel has completed `9 / 3 = 3` revolutions.  
+
+The small wheel, however, has a circumference one quarter the size, so `3/4`m or 75cm.  
+
+900cm / 75cm = 12 revolutions.  
+
+So in total the wheels have turned 3 + 12 = 15 revolutions.
+:::
+
+
+## 2. Going round in circles.
+
+::: problem id=2_1 marks=2
+__2.1__ In the diagram, the smaller circle touches the larger circle and also passes through its centre.
+
+![](/resources/6-16-wobbly-circles/2-circles.png){image align="center"}
+
+What fraction of the area of the larger circle is the smaller circle?  
+
+* [ ] The small circle is half the area of the large
+* [ ] The small circle is one third the area of the large
+* [x] The small circle is one quarter the area of the large
+* [ ] The small circle is one sixth the are of the large
+
+---
+
+First think of a square: if we half the length, what happens to the area?  
+
+A 6cm square has an area of 36cm<sup>2</sup>, if we half that length we get a 3cm square with area 9cm<sup>2</sup>.  
+
+So the area has been quartered!
+
+We can apply this to the circle too, if we half the diameter we will quareter the area (because it is both half the width AND the height of the original circle).  
+
+You might also be able to look at this dagram and convince yourself (correctly) that the shaded area is equivalent to the area of the smaller circles.  
+
+![](/resources/6-16-wobbly-circles/2-circles-solution.png){image align="center"}
+:::
+
+
+Before you hit the SUBMIT button, here are some quick reminders:
+
+*	You will receive your score immediately, and collect your reward points.
+*	You might earn a new badge... if not, then maybe next week.
+*	Make sure you go through the solution sheet – it is massively important.
+*	A score of less than 50% is ok – it means you can learn lots from your mistakes.
+*	The next Parallelogram is next week, at 3pm on Thursday.
+*	Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.
+
+Cheerio,
+Simon and Ayliean.

pages/6-17-coin-quandaries.md

@@ -0,0 +1,122 @@
+# Coin quandaries
+
+<div class="dictionary">
+
+__Noun__: Parallelogram
+__Pronunciation__: /ˌparəˈlɛləɡram/
+
+1. a portmanteau word combining parallel and telegram. A message sent each
+week by the Parallel Project to bright young mathematicians.
+
+</div>
+
+*	Tackle each Parallelogram in one go. Don’t get distracted.
+*	Finish by midnight on Sunday if your whole class is doing parallelograms.
+*	Your score & answer sheet will appear immediately after you hit SUBMIT.
+*	Don’t worry if you score less than 50%, because it means you will learn something new when you __check the solutions__.
+
+
+## 1. The Hexacoin Puzzle
+
+This puzzle seems impossible, but Alex Belos shows that it can be done - although it is easy to forget how!  
+
+Watch this video, remember to pause and give it a go.  
+
+ @[youtube](_pP_C7HEy3g) _(If you have problems watching the video, right click to open it in a new window)_
+
+::: problem id=1_0 points=1
+__1.0__ Honesty point: if you had a go at the coin puzzle, whether you completed it or not, click here for one point!
+* [x] I tried!
+:::
+
+::: problem id=1_1 marks=2
+__1.1__ When the coins are moved, what rule must they follow?  
+
+* [ ] The can only be touching one other coin
+* [x] They must be touching two other coins
+* [ ] They must be touching three other coins
+:::
+
+::: problem id=1_2 marks=2
+__1.2__ Using the standard UK amounts of 1p, 2p, 5p and 10p, how many different ways are there to make 10p?  
+
+<input type="number" solution="11"/>
+
+---
+
+There are many logical ways to approach this. One would be to consider that there is only one solution using the 10p.  
+
+There are 10 other solutions using combinations on 1p, 2p, and 5p coins.  
+
+As shown in this venn diagram:  
+
+![](/resources/6-17-coin-quandaries/1-coin-venn.png){image align="center"}
+:::
+
+::: problem id=1_3 marks=2
+__1.3__ You have 15 one pound coins, which you divide into 4 money bags.  
+
+The coins are divided such that you can hand over any amount from £1 to £15 without opening a bag, just handing over a combination of money bags.  
+
+How are coins divided?
+
+* [ ] £1, £3, £5, £6
+* [x] £1, £2, £4, £8
+* [ ] £2, £3, £4, £6
+* [ ] £1, £1, £6, £7
+:::
+
+
+## 2. When life gives you (puzzles about) lemons
+
+::: problem id=2_1 marks=2
+__2.1__ Oranges cost 26p each and lemons cost 30p each.  
+
+Gordon spent exactly £5 on a mixture of oranges and lemons.  
+
+How many oranges and lemons altogether did he buy?  
+
+* [ ] 16
+* [ ] 17
+* [x] 18
+* [ ] 19
+* [ ] 20
+{.col-5}
+
+^^^ hint id=2_1_1 marks=1
+£5 is a multiple of 10, the cost of a lemon is also a multiple of 10.  
+
+This means we need to buy enough oranges to make the total price of the oranges a multiple of 10.
+^^^
+
+^^^ hint id=2_1_2 marks=1
+If the cost of the oranges must be a multiple of 10, then we have bought either 10 or 20 oranges.  
+
+For a total of £5, only one of these is possible.
+^^^
+
+---
+
+£5 is a multiple of 10, the cost of a lemon is also a multiple of 10.  
+
+This means we need to buy enough oranges to make the total price of the oranges a multiple of 10.
+
+If the cost of the oranges must be a multiple of 10, then we have bought either 10 or 20 oranges.  
+
+20 oranges would cost £5.20, which is too much!
+
+So we have 10 oranges at a cost of £2.60 and then with the remaining £2.40 we can buy 8 lemons, giving us a grand total of 18 fruits all together.  
+:::
+
+
+Before you hit the SUBMIT button, here are some quick reminders:
+
+*	You will receive your score immediately, and collect your reward points.
+*	You might earn a new badge... if not, then maybe next week.
+*	Make sure you go through the solution sheet – it is massively important.
+*	A score of less than 50% is ok – it means you can learn lots from your mistakes.
+*	The next Parallelogram is next week, at 3pm on Thursday.
+*	Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.
+
+Cheerio,
+Simon and Ayliean.