Parallelogram 37 • Level 3 • 16 May 2024Mathematical Timeline

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

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:

2 marks

1.1 Looking at mathematicians from 1000 to 1500, what is the surname of the mathematician who invented co-ordinate geometry?

Correct Solution: ORESME

2 marks

1.2 Looking at mathematicians from 1000 to 1500, what is the first name of the mathematician who used an infinite series to work out an exact value for π?

Correct Solution: MADHAVA

2 marks

1.3 Looking at mathematicians from 1500 to 1600, what is the surname of the mathematician who helped discover (or maybe invent) imaginary numbers, as well as working on everything from locks to gyroscopes? (Clue: initials GC)

Correct Solution: CARDANO

2. Junior Maths Challenge Problem (UKMT)

2 marks

2.1 A train carriage has 80 seats. On my journey I noticed that all the seats in my carriage were taken and 7 people were standing.

At Banbury, 9 people left the carriage, 28 people entered it and all the seats were taken.

How many people now had no seat?

0
7
16
26
35
(Not answered)

At Banbury 9 people left the carriage and 28 entered it. Therefore after the stop at Banbury there were 28 − 9, that is, 19, more people in the train carriage than before. Therefore, as all the seats were taken, the number of people who were standing went up from 7 to 7 + 19, that is, to 26.

Therefore there were now 26 people who had no seat.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 I add up all even numbers between 1 and 101. Then from my total I subtract all odd numbers between 0 and 100. What is the result?

0
50
100
255
2525
(Not answered)

The sum we are asked to work out is:

(2 + 4 + 6 + ... + 96 + 98 + 100) − (1 + 3 + 5 + ... + 95 + 97 + 99),

where ... indicates that we have left out some numbers because of lack of space. We may rewrite this expression as:

(2 − 1) + (4 − 3) + (6 − 5) + ... + (96 − 95) + (98 − 97) + (100 − 99),

which is equivalent to

1 + 1 + 1 + ... + 1 + 1 + 1.

There are fifty 1s in this expression, and hence the sum is 50.

4. Junior Maths Challenge Problem (UKMT)

4 marks

4.1 The small trapezium on the right has three equal sides and angles of 60° and 120°.

Nine copies of this trapezium can be placed together to make a larger version of it, as shown.

The larger trapezium has perimeter 18 cm.

What is the perimeter of the smaller trapezium?

2 cm
4 cm
6 cm
8 cm
9 cm
(Not answered)

Show Hint (–1 mark)

–1 mark

The large trapezium and the small trapezium are made from three equilateral triangles. The longer side is twice as long as any of the shorter sides.

Large trapezium – the total perimeter (18 cm) is equal to (5 x any of the shorter sides), which means any of the shorter side is 3.6 cm in length, and the longer side is 7.2 cm.

Small trapezium – the longer side and one shorter side (which are equivalent in length to 3 shorter sides) are equal to 3.6 cm, so each short side is 1.2 cm.

Therefore, the perimeter of the small trapezium is 5 x 1.2 cm = 6 cm.

5. Amazing James Randi and psychic surgery

A couple of weeks ago, you watched the Amazing James Randi perform some escapology. Today, he is back again, this time performing some psychic surgery:

This video looks gruesome, so don’t watch it if you are squeamish.
Psychic surgery is fake, but some people believe it is real.
People waste time, money and hope visiting psychic surgeons.
See if you can work out how Randi performs his fake surgery.
Really, don’t watch it if you are squeamish!

(If you have problems watching the video, right click to open it in a new window)

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.

Additional Stuff

Here is another clip featuring the Amazing James Randi, in which he tests someone who claims he can move objects with the power of his mind. I think this clip is terrific because it shows how a bit of clever thinking can create a test that demonstrates that this psychic has no special powers. First you will see the fake psychic and then your will see Randi’s clever test. (I should add that no psychic has ever convinced the scientific community that they have special superhuman powers, so the assumption is that all psychics are fake.)