Parallelogram 22 • Level 2 • 1 Feb 2024It's all downhill from here

• 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.

This Parallelogram first appeared at the time of the Beijing Winter Olympics, so it has a winter sports theme!

1. Winter Olympics Puzzle (A)

Every four years, the world’s leading winter athletes – the skiers, skaters, sledders and snowboarders – gather to compete for medals in the Winter Olympic Games.

Of course, among the drama, bravery and triumph, there is plenty of room for maths! So, while you’re cheering on your favourites, don’t miss the chance to tackle these Winter Olympics maths puzzles.

The Jamaican bobsled team was one of the most famous feel-good stories of the 1988 Games in Calgary. Jamaica’s Caribbean climate is not the most helpful for bobsledding, but the team borrowed equipment and asked for help. While the four-man bob team was unplaced after crashing in qualifying, the two-man team finished a creditable 30th out of 41. The movie Cool Runnings is – very loosely – based on their experience.

In the bobsled (or bobsleigh), teams of two or four pack into a streamlined sled and slide down a twisting, icy course with steep, banked corners as fast as they possibly can – often as fast as 75mph. It gets its name from early competitors who realised they could go faster by bobbing back and forth inside the sled!

The 2010 Winter Olympics at Whistler, Canada, took place on a track 1,450 metres long. Each team had four runs at the course, and the winners were the team with the smallest total time.

Feel free to use a calculator for these puzzles!

2. Winter Olympics Puzzle (B)

The most famous ski jumper in history is not exactly famous for being good. Eddie “The Eagle” Edwards, self-funded and short-sighted, famously finished in last place in both the 70m and 90m events at Calgary, Canada in 1988, scoring fewer than half as many points as his nearest competitor. However, these scores are still the British records for ski-jumping! In 2016, a film of his exploits, Eddie The Eagle, was released.

In the ski jump, competitors ski down a huge ramp, reaching speeds of up to 60mph before jumping as far down the hill as they possibly can.

Each jump is awarded points based on two factors: the distance jumped, and the jumper’s style.

Here is the recipe (we've simplified it a bit for this puzzle!):

• Take 60 metres away from the distance jumped (this could be a negative number!)
• Double the result
• Add on the style score.

A group of judges awards up to 60 points for style – how well the jumper flies and lands.

For example, to work out the score for a ski jumper who jumped 95 metres with enough grace to pick up 40 style points, you would calculate:

95 – 60 = 35
35 × 2 = 70
70 + 40 = 110

...so the jump would be worth 110 points.

Similarly, a jumper who manages 80 metres with 25 style points would score 65 points altogether.

This puzzle and the previous one come from the Komodo website, which offers a programme of maths, largely aimed at younger children. I’ve used it with my son, and I think it offers a great way to practice core arithmetic skills.

The puzzles were created by Colin Beveridge - a maths tutor, former NASA researcher, and writer of maths books including Basic Maths for Dummies.

3. The physics of ski jumping

To understand the underlying physics in the previous question, take a look at this video that explains how to be a ski jump champion... and then answer this question.

4. Junior Maths Challenge Problem (UKMT)

The diagram shows a cuboid in which the area of the shaded face is one-quarter of the area of each of the two visible unshaded faces. The total surface area of the cuboid is 72 cm².

5. The affine cipher

Over the last couple of weeks, we discussed how you can encode a message by turning each letter into a number according to the alphabet-number pairings below, and either (1) adding a fixed value to each letter-number or (2) multiplying each letter-number by a fixed value. The result is then turned back into a letter.

The affine cipher uses both multiplication and addition to encipher a letter, word or whole message. For example, let’s multiply each letter in the message (L_M) by 5 and then add 2 to get each letter in the code (L_C). We could say:

L_C = (5 × L_M) + 2

Let’s apply this affine cipher to the word CAB. If we turn this into numbers, we obtain 2 ¦ 0 ¦ 1.

If we multiply by 5 and add 2, we obtain 12 ¦ 2 ¦ 7.

If we translate back into letters, the code is MCH.

Now, let’s encipher the same word, but this time we will change the affine cipher so that we multiply by 9 and add 11:

L_C = (9 × L_M) + 11

Again we will start with the word CAB. If we turn this into numbers, we obtain 2 ¦ 0 ¦ 1.

If we multiply by 9 and add 11, we obtain 29 ¦ 11 ¦ 20.

Remember, we cannot have numbers bigger than 25, so we subtract 26 from any problematic numbers, so that 29 becomes 3. So, the new letter sequence is 3 ¦ 11 ¦ 20.

If we translate back into letters, the code is DLU.

If you are confused (perhaps you missed some of the earlier Parallelograms), then take a look at this video from Chris at Chegg.com. The explanation is fairly slow and lengthy, but it should clear up any confusion.

Now that you have mastered the affine cipher, answer these questions.

6. Talking maths

Use this article from Merriam-Webster.com to match each word with its literal meaning.

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.