Parallelogram 48 • Level 4 • 31 Jul 2025Perplexing paperclips
Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/
- a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
- Tackle each Parallelogram in one go. Don’t get distracted.
- 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.
It’s summer and the school year is over (or maybe it’s nearly over), but that doesn’t mean that it’s the end of mathematics. If you are taking mathematics seriously and if you are having fun with it (being serious and having fun can go together), then it is important that you keep the numbers and geometry parts of your brain at least a little bit busy over the summer. I am hoping that your teacher will have set you some holiday work, but if you have missed any earlier Parallelograms then the summer is a great time to do some catching up.
1. The amazing Tadashi and perplexing paperclips
Tadashi Tokieda is one of my favourite mathematicians. He is one of the most curious people in the world, finding wonder and joy in the most ordinary of objects. I have sprinkled some Tadashi videos through this Parallelogram, and to start with here is a video about paperclips.
(If you have problems watching the video, right click to open it in a new window)1.1 According to Tadashi, in terms of learning and advancing science, “it’s always important to...”
- smile
- guess
- be polite
- eat your greens
- stick your chewing gum on your bedpost
- (Not answered)
2.
2.1. What is the value of
- 540
- 720
- 900
- 1080
- 1440
- (Not answered)
Remember, for every polygon the sum of its exterior angles is 360°.
We use the fact that for every polygon the sum of its exterior angles is 360°.
The shaded angles in the diagram on the right form one set of external angles. Therefore
The other marked angles form another set of external angles and therefore
[Alternatively, we could argue that the angles marked
It follows that
3.
3.1 An ant is on the square marked with a black dot.
The ant moves across an edge from one square to an adjacent square four times and then stops.
How many of the possible finishing squares are black?
- 0
- 2
- 4
- 6
- 8
- (Not answered)
The crucial point with this question is that we are not dealing with a flat board, but rather a board with a step, and this means it is possible move from a white square to a white square. So you can reach a black square by moving from white to black to white to white to black.
The figure shows the part of the layout that is relevant. Some of the squares have been labelled so that we may refer to them. A movement of the ant will be indicated by the sequence of labels of the four squares, in order, that the ant travels to after leaving the square labelled P. For example, QTUV is one such sequence. In this example the ant ends up on the black square V.
The ant starts on a white square. If at each move the colour of the square it is on changes, after four moves it will end up on a white square. So the only way the ant can end up on a black square is if on one of its moves the colour of the square does not change. That is, during its sequence of moves it either moves from T to U, or from U to T, or from W to X, or from X to W.
We therefore see that the only movements which result in the ant ending up on a black square are
QTUR, QTUV, QTUX, QTWX, RUTQ, RUTS, RUTW and RUXW.
From the final letters of these sequences we see that there are exactly 6 black squares that the ant can end up on after four moves, namely, Q, R, S, V, W and X.
4.
4.1. The total weight of a box, 20 plates and 30 cups is 4.8 kg. The total weight of the box, 40 plates and 50 cups is 8.4 kg.
What is the total weight of the box, 10 plates and 20 cups?
- 3 kg
- 3.2 kg
- 3.6 kg
- 4 kg
- 4.2 kg
- (Not answered)
The difference between the box, 20 plates, 30 cups and the box, 40 plates, 50 cups, is 20 plates and 20 cups. From the information in the question we see that the extra weight of 20 plates and 20 cups is 8.4 kg − 4.8 kg = 3.6 kg.
Therefore the weight of 10 plates and 10 cups is
If we remove 10 plates and 10 cups from the box, 20 plates and 30 cups we end up with the box, 10 plates and 20 cups. Removing the 10 plates and 10 cups reduces the original weight of 4.8 kg by 1.8 kg. Hence the weight of the box, 10 plates and 20 cups is 4.8 kg − 1.8 kg = 3 kg.
5.
5.1 There are 120 different ways of arranging the letters, U, K, M, I and C. All of these arrangements are listed in dictionary order, starting with CIKMU.
Which position in the list does UKIMC occupy?
- 110th
- 112th
- 114th
- 116th
- 118th
- (Not answered)
Because the sequence UKIMC comes towards the bottom of the list in dictionary order of all 120 arrangements of the letters U, K, M, I, C, we can most easily find its position by listing the arrangements in reverse dictionary order, until we reach UKIMC, as follows
120 UMKIC
119 UMKCI
118 UMIKC
117 UMICK
116 UMCKI
115 UMCIK
114 UKMIC
113 UKMCI
112 UKIMC
We see from this that UKIMC is the 112th arrangement in the dictionary order list.
And that’s it for this academic year. I hope you’ve enjoyed doing Parallel this year and make sure you come back in September, when we will have another year of Parallelograms.
But, before that, a few important points.
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...
- 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.
- 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.
- This was our last Parallelogram of the year, but be sure to come back in September for more puzzles and problems.
Cheerio, Simon.