Parallelogram 38 • Level 2 • 23 May 2024Joseph's Machines
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. History of Mathematics
As mentioned last week, the terrific Mathigon website has loads of great material, including a timeline showing the great mathematicians of past centuries. Visit the timeline (click and it will open up in a new tab) and answer the three questions below. Just enter the name given in the plum box.
So, if the answer is John Napier, just enter Napier, because that is the name that appears in the plum-coloured box on the timeline:
1.1 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who invented the Game of Life? (Clue: initials JHC)
Correct Solution: CONWAY
1.2 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the Iranian-born mathematician who won the highest honour in mathematics, the Fields Medal? (Clue: initials MM)
Correct Solution: MIRZAKHANI
2. Junior Maths Challenge Problem (UKMT)
2.1 The positive integers from 1 to 150 inclusive are placed in a 10 by 15 grid so that each cell contains exactly one integer. Then the multiples of 3 are given a red mark, the multiples of 5 are given a blue mark, and the multiples of 7 are given a green mark. How many cells have more than 1 mark?
- 10
- 12
- 15
- 18
- 19
- (Not answered)
The squares that receive more than one mark are those which contain integers which are divisible by at least two of the integers 3, 5 and 7. The integers 3 and 5 are coprime, that is, they have no common factor other than 1. Therefore the integers that are divisible by 3 and 5 are precisely those that are divisible by 3 × 5, that is, by 15. Every 15th integer is divisible by 15. Hence the number of integers in the range from 1 to 150 that are divisible by 15 is (
Similarly, the integers that are divisible by 3 and 7 are precisely those that
are divisible by
Similarly, the integers that are divisible by 5 and 7 are precisely those that are divisible by
So it seems that we have
However, you need to check for numbers that are divisible by 3, 5 and 7, because they would have been counted three times, not just once – when you look for numbers that are divisible by 3 and 5, then 3 and 7, then 5 and 7. In fact, the only number less than 150 divisible by 3, 5 and 7 is 105, so you only need to remove 2 from your tally, so that it is only counted once, i.e., the answer is
3. Sumaze! Part 1
Sumaze! is a terrific game that tests your core maths skills. It was created by Richard Lissaman, who is part of MEI (Mathematics in Education & Industry).
You will have access to 9 levels of the game, starting with some really trivial ones to allow you to understand the game, followed by some trickier levels. You score 1 point for level 1, 2 points for level 2, 3 points for level 3 and so on up to 9 points for level 9.
However, you will lose 2 points for every hint you use (available on levels 8 & 9). And if you have to restart a level then you will lose 1 point each time, up to a maximum of 4 lost points. So, if you need one hint on level 9, then your maximum score goes from 9 down to 7. And if you succeed after restarting three times, then your score drops from 7 down to 4.
It is easier to understand Sumaze! by playing it, but in case it helps here are a few important rules and tips:
- Aim to move your blue box to reach the “end of level” square.
- Your box cannot at any time contain a number bigger than 1,000.
- Your box cannot at any time contain a number that is a fraction.
- You sometimes need keys to open doors to reach the “end of level” square.
- Passing through an arrow means you cannot go back.
- To pass through a green square you must satisfy the square’s condition.
- Pink squares (and their operations) disappear when you move onto them.
- Lilac squares (and their operations) can be crossed over and over again.
- IMPORTANT – Sumaze won't run on a smartphone. If you are currently on a smartphone, then type this into your PC/Mac/tablet browser: bit.ly/parallel_sumaze
You can stop and start Sumaze!, and the website will use cookies to remember your level and score so far, but it is better to finish all the levels once you have started.
3.1 When you have finished Sumaze, you will receive a score and a code word. Please enter your code word here, so that we can add your Sumaze score to your overall Parallelogram total.
Score: {{c.sumaze(c.answers.p_3_0)}} / 45
The maximum score in Sumaze is 1 + 2 + 3 + … + 9 = 45. We will divide your Sumaze score by 10 and add it to your overall Parallelogram score. So ultimately a perfect Sumaze score will add 45/10 = 4.5 to your Parallelogram total score.
This video clip shows the paths you should or could have taken to solve each stage with a perfect score.
4. The Page Turner
If you scroll down, you will see one of my favourite YouTube videos, which features a complex machine designed just to turn the page of a newspaper. It is the work of the eccentric inventor Joseph Herscher. While you watch it, try to answer the questions below.
4.1 The 4-ball encounters three goblets, but it does not drop into any of them because they are filled with:
- Sand
- Balls
- Water
- Cheese
- Compressed air
- (Not answered)
4.2 How does the water move from a beaker to a sponge, causing the sponge to get heavier and tip down?
- Boiling
- Freezing
- Pipe
- Pipette
- Suction
- (Not answered)
5. Sumaze! Part 2
After the earlier taster, it's time for another set of Sumaze! challenges. The rules are the same, and you have 9 more levels to puzzle your way through!
5.1 When you have finished Sumaze, you will receive a score and a code word. Please enter your code word here, so that we can add your Sumaze score to your overall Parallelogram total.
Score: {{c.sumaze(c.answers.p_2)}} / 45
The maximum score in Sumaze is 1 + 2 + 3 + … + 9 = 45. We will divide your Sumaze score by 10 and add it to your overall Parallelogram score. So ultimately a perfect Sumaze score will add 45/10 = 4.5 to your Parallelogram total score.
This video clip shows the paths you should or could have taken to solve each stage with a perfect score.
6. Another of Joseph’s machines – his longest machine ever
6.1 Another amazing invention by Joseph Herscher. Just one question. How many ping pong balls are tipped onto the table in order to just knock over a couple of dominos?
Correct Solution: 12
7. Parallel emails
If you use Parallel on an email address from your school, some school IT systems prevent you from receiving the emails we send out to remind you when a new Parallelogram is released, or when we have another exciting and nerdy maths thing to tell you about. To avoid this, you could give us a different email address we can contact you on - either a personal email address, or one for you parents or guardian. If you'd like to do that, you can put it in this form, but please ask a parent first if you are not yet 13 years old.
7.1 If you do want to give us an alternate email address, be sure to click the link above (if you haven’t already). Either way, here's a free mark, just for being a fan of Parallel.
- Thank you!
- (Not answered)
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.
- 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.
- The next Parallelogram will be out next Thursday at 3pm.
Cheerio, Simon.