Parallelogram 28 • Level 5 • 14 Mar 2024Magic Hexagons

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.
When you finish, remember to hit the SUBMIT button.
Finish by Sunday night if your whole class is doing parallelograms.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Magic Hexagon

Take a look at this video by James Grime and answer the question below.

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

2 marks

1.1 How many types of magic hexagon are there?

0
1 (with 19 elements)
2 (with 1 or 19 elements)
3 (with 1, 2 or 3 elements)
∞
(Not answered)

2.

3 marks

2.1 What is the value of 20032?

4009
400009
401209
4000009
4012009
(Not answered)

2000+32=20002+2×2000×3+32=4000000+12000+9=4012009.

3.

4 marks

3.1 Lines AB and CD are parallel and BC=BD.

Given that x is an acute angle not equal to 60°, how many other angles in this diagram are equal to x?

1
2
3
4
5
(Not answered)

Show Hint (–1 mark)

–1 mark

The answer is not 2.

Angle BCD = x (vertically opposite angles);
Angle BDC = angle BCD (base angles of isosceles triangle);
Angle ABC = angle BCD (alternate angles);
Angle EBD = angle BCD (alternate angles).

4.

4 marks

4.1 Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?

90
179
180
359
360
(Not answered)

Show Hint (–1 mark)

–1 mark

In order for the polygon to have the largest number of sides, the internal angle needs to be as large as possible, which is 179°, as the angle has to be a whole number and it cannot be 180° or larger.

As each interior angle of the polygon is a whole number of degrees, the same must apply to each exterior angle. The sum of the exterior angles of a polygon is 360° and so the greatest number of sides will be that of a 360-sided polygon in which each interior angle is 179°, thus making each exterior angle 1°.

5.

4 marks

5.1 When dates are written using eight digits, eg 06 02 2020, the 20th February 2002 is a palindromic date, since 20 02 2002 has the same digits in the same order when read in reverse.

The previous palindromic date and the next few all occur in the month of February.

What will be the next month other than February to have a palindromic date in it?

March
April
October
November
December
(Not answered)

All palindromic dates this century will fall in February as they will be of the form ab 02 20ba.

The palindromic dates next century will fall in December as they will be of the form cd 12 21dc and the first of these will be the 10th December 2101, ie 10 12 2101.

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.