Parallelogram 13 Level 1 30 Nov 2023Alright for sum

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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. The square sum problem

Watch this video where famously square person Matt Parker tries to solve this square sum puzzle.

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

You can just watch the video - don't worry about pausing to give Matt’s puzzle a go here, because it's quite difficult!

2 marks

1.1 Which of these sums does not add to give a square number?

  • 1 + 9
  • 1 + 15
  • 2 + 7
  • 30 + 6
  • (Not answered)

Square numbers are the result of multiplying a number by itself.

1 + 15 = 16 and 16=4×4 aka 42.

The only option on this list which does not have this property is 1 + 9 = 10. 10 is a nice number, but it is not square.

1 mark

1.2 Matt’s puzzle only used the numbers 1 to 15, would it have been possible to do with the numbers 1 to 10

  • Yes, it is possible for any amount of numbers
  • No, it only works for the numbers 1 to 15
  • No, 15 is the first point where the puzzle starts to work
  • (Not answered)

Only once we have the numbers 1 to 15 it is possible to take a path along every point in the graph.

There are other amounts greater than 15 where the path will work again.

1 mark

1.3 What is the name of the path which goes through every point on a path?

  • Garden Path
  • Hamiltonian Path
  • Washingtonian Path
  • Square Path
  • (Not answered)
2 marks

1.4 Is there a path to pass through every corner on a cube without going along the same edge twice?

  • Yes
  • No
  • It depends on the size of the cube
  • (Not answered)

There is a Hamiltonian Path along the edges of a cube.

In fact there is a Hamiltonian Path along the edges of all of the Platonic Solids!

The 3D world is a good place to be.

2 marks

1.5 Look at these pairs of numbers. For each pair, square each number and add the results.

When you do this, which pair gives another square number?

  • 7 and 10
  • 3 and 4
  • 3 and 9
  • 1 and 2
  • (Not answered)

2. A puzzle adapted from the UKMT challenge

3 marks

2.1 I have a list of 6 consecutive numbers.

If the sum of all the even numbers in my list is 24.

What is the smallest possible total sum of every number on my list?

  • 45
  • 48
  • 50
  • 52
  • (Not answered)
Show Hint (–1 mark)
–1 mark

If the list contains 6 numbers there must be three even, and three odd numbers.

The odd numbers could be one more or one less than their even counterparts on the list, consider what happens in each case.

There are two possible solutions to this question.

Either the list starts on an even number in which case each of the three odd numbers is one more than their even counterpart. The sum of the odd numbers in this case would be 24 + 3.

This gives a total sum of 24 + 24 + 3 = 51 (which is not an option.

So we know that the list must start on an odd number. In this case each of the odd numbers is one less than its even counterpart on the list.

Giving an odd total of 24 - 3.

This gives us an overall list total sum of 24 + 24 - 3 = 45.

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 and Ayliean.