Parallelogram 44 • Level 1 • 3 Jul 2025Passcode logic
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.
Many problems seem impossible at first because they do not appear to have much information.
But with the power of logic, we can deduce our way to the answer by creatively combining the bits of information we do have.
1. Passcode logic
In this video you are asked to solve a riddle in which you must figure out a passcode.
Make sure you pause the video before the solution is revealed and have a think through the clues provided.
With some clever logic, you will find the solution without any guesswork at all.
(If you have problems watching the video, right click to open it in a new window)1.1 Going on the first clue alone, which of these options will definitely not work?
- 1-6-6
- 3-3-4
- 3-4-5
- 1-2-18
- (Not answered)
The first clue tells us that the three numbers must multiply to 36, but 3 × 4 × 5 = 60.
1.2 Why doesn’t Zara solve the puzzle after the second clue?
- She forgot her hallway number
- She does not have enough information to be certain of the answer
- She can’t be bothered
- She knows the answer but by waiting for more clues she gives you more information
- (Not answered)
Zara is a perfect logician, so if she had enough information she would not hesitate to solve the puzzle.
1.3 If the third clue told us that the smallest number is repeated, what would Zara conclude?
- The passcode is 1-6-6
- The passcode is 1-1-36
- The passcode is 2-2-9
- She would not be able to deduce the passcode
- (Not answered)
The two options by this point are 1-6-6 and 2-2-9.
The two options by this point are 1-6-6 and 2-2-9 and since the smallest number is repeated, the code must be 2-2-9.
1.4 If the third clue told us that the passcode contains exactly one square number, what would Zara conclude?
- The passcode is 1-1-36
- The passcode is 2-2-9
- The passcode is 3-3-4
- She would not be able to deduce the passcode
- (Not answered)
The two options by this point are 1-6-6 and 2-2-9
The two options by this point are 1-6-6 and 2-2-9.
Both options contain one square number (1 and 9 respectively) so the correct code can not be deduced.
2. Square tiles and square numbers
2.1 This pattern is made from 25 small square tiles.
How many different kinds of tile are there?
- 2
- 3
- 4
- 5
- 6
- (Not answered)
There are four distinct tiles:
2.2 Who am I? I am the sum of the squares of two consecutive numbers less than 20, and I look the same upside down!
- 101
- 161
- 181
- 808
- 818
- (Not answered)
The answer is the sum of an odd number and an even number, so it must be odd.
The answer is the sum of an odd number and an even number, so it must be odd. So we can rule out 808 and 818.
We can also rule out 161 as it does not look the same upside down.
Of the remaining two options, only 181 works, and it is given by
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.