Parallelogram 46 • Level 1 • 17 Jul 2025Any number any card trick
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.
- 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.
Many card tricks can be explained with a bit of maths.
In this video you will see the trick performed.
Watch through to the end to see how the deck must be prepared beforehand.
We will then get into the explanation of why it works.
(If you have problems watching the video, right click to open it in a new window)1.1 If the spectator chooses 35, what card does this correspond to?
- 3 of Clubs
- 8 of Clubs
- 8 of Spades
- 3 of Hearts
- (Not answered)
For the tens digit, 1 = Spades, 2 = Hearts, 3 = Clubs, 4 = Diamonds.
The sum of the two digits gives the value of the card (3 + 5 = 8 in this case).
1.2 If a spectator’s card turns out to be the 5 of Hearts, what number must they have chosen?
- 5
- 23
- 32
- Impossible to determine
- (Not answered)
Since it is a Hearts, the tens digit is 2.
Since the digits add up to 5, the Ones digit must therefore be 3.
1.3 Which of these cards can NOT be revealed in this trick?
- 3 of Spades
- 3 of Clubs
- 3 of Diamonds
- Every card can be revealed depending on the number chosen
- (Not answered)
If the spectator chooses a Diamonds then their tens digit is a 4.
If the spectator chooses a Diamonds then their tens digit is a 4 and the value of the card, which is the sum of their two digits, already exceeds 3.
So 3 of Diamonds is impossible.
1.4 The trick works because the cards are set up in a particular way.
First you create 4 piles - one for each of the 4 suits (Spades, Hearts, Clubs, Diamonds).
How many cards are in these 4 piles?
- 10 each
- 13 each
- 8, 9, 10, 11 respectively
- 9, 10, 11, 12 respectively
- (Not answered)
The piles consist of:
- Spades: Ace to 10
- Hearts: 2 to Jack
- Clubs: 3 to Queen
- Diamonds: 4 to King
Each suit corresponds to a range of 10 numbers (10-19 for Spades, 20-29 for Hearts, 30-39 for Clubs, 40-49 for Diamonds), so we need 10 cards for each of the 4 piles.
They are:
- Spades: Ace to 10
- Hearts: 2 to Jack
- Clubs: 3 to Queen
- Diamonds: 4 to King
2. Tricky problems
2.1 The diagram shows a pentagon with a square placed edge to edge with it.
George places another square on each of the other four edges of the pentagon.
How many outside edges does the combined shape now have?
- 13
- 14
- 15
- 16
- 17
- (Not answered)
Each square contributes three edges to the resulting shape, so the polygon will have 3 × 5 = 15 edges.
2.2 A ten foot pole casts an eight foot shadow.
How long is a pole which casts a twelve foot shadow, in feet?
Correct Solution: 15
The height of an object is directly proportional to its shadow, and the proportion is
The height of an object is directly proportional to its shadow, and the proportion is
So the length of the pole is
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.