Parallelogram 30 • Level 1 • 28 Mar 2024Mostly magic squares
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. The Parker Square
Magic squares have fascinated people throughout history, but mathematicians still don’t know everything about them!
Watch this video where Matt Parker tries his best at this unsolved magic square problem
(If you have problems watching the video, right click to open it in a new window)1.1 In this magic square the rows, columns and diagonals add to the same magic total.
Which number should be in the top right?
Correct Solution: 5
The magic constant is 3 + 6 + 9 = 18 so the missing number in the top right is 18 - (10 + 3) = 5
1.2 This is a multiplication magic square, all of the rows, columns and diagonals multiply to the same magic total.
Which number should be in the top middle?
Correct Solution: 1
The magic constant is 216, this is the smallest possible constant for a multiplication magic square.
18 × 12 = 216 so the missing number must be 1.
1.3 There is nothing to stop magic squares including negative numbers.
What is the magic total for this addition magic square?
Correct Solution: -9
2. Gridlock
2.1 In the multiplication grid shown, the input factors (the ones that would appear in the first row and the first column) are all missing.
Only some of the products within the table have been given.
What is the value of A + B + C + D + E?
- 132
- 145
- 161
- 178
- 193
- (Not answered)
In column A we have 15 and 18 - what times table includes both of these numbers? This should be the input factor and go at the top of the column.
Now we can figure out row C.
In row B we have 15 and 40, and they're both in the 5 times table! Put 5 as the input factor at the start of the row.
Now we can figure out column C.
The 2nd row must be multiples of 5, as that is the only common factor of 15 and 40.
So 15 is 5 × 3 and 40 is 5 × 8.
Therefore, we know that the first column is headed by 3 and the middle column is headed by 8.
In turn, this tells us that the input factor for the bottom row is 7, because 7 × 8 = 56.
Eventually, we can deduce that A + B + C + D + E = 6 + 25 + 48 + 40 + 42 = 161
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.