Parallelogram 12 • Level 2 • 23 Nov 2023KenKen Level 2
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. KenKen
Today’s Parallelogram is dedicated to KenKen – a mathematical puzzle that is ten times more interesting than Sudoku.
First of all, these are the rules.
- You will be presented with a 3 x 3 grid.
- Each row and each column contain the numbers 1, 2 and 3 once and only once.
- Therefore, no number can be repeated in a row or column.
- Each marked region (or cage) contains a target number and a mathematical operation. The numbers in the cage must add up, subtract, multiply or divide to give the target number.
- The ordering of the numbers does not matter in a cage. So, 2 followed by 1 or 1 followed by 2 can both give the target (1,−), which means reaching the target 1 by subtraction.
- If the cage contains just 1 cell, the value of the cell is the same as the target number. it is a “freebie”.
- There are also 4 x 4 grids, and each row contains the numbers 1, 2, 3 and 4 once and only once. There are also even bigger grids.
This video summarises the rules of KenKen.
If you’re still unsure, this video goes through the rules one more time.
(If you have problems watching the video, right click to open it in a new window)1.1. Here is your first KenKen. The only mathematical operation is addition.
You can either copy it onto a bit of paper by hand or follow this link to open a copy of all of the questions that you can print and fill in.
To help us check whether you have completed the grid correctly, add up the 4 corner numbers and enter the sum in the box below.
The (4 +) cage is one way to start cracking this KenKen. The cage has to contain (3,1), because the only other way to obtain 4 is (2, 2) and no number can be repeated in a row. We don’t know the order of the (3, 1) cage, but it tells us the remaining number in that column must be (2).
Correct Solution: 8
2. Two more KenKens
2.1. What do the four corner cells add up to?
The (3 +) cage has to contain (1,2), because it is the only way to obtain 3. We don’t know the order of the (1, 2) cage, but it tells us the remaining number in that column must be (3).
Correct Solution: 9
2.2. This KenKen involves multiplication and division.
What do the four corner cells add up to?
The (18x) cage has to contain (1,2, 3, 3), because it is the only way to obtain 18 by multiplying the four allowable digits. The repeated 3 is helpful, because they cannot both be in the top row, so one of them must be in the right column, middle row square.
Correct Solution: 8
3. Two tougher KenKens
3.1 This KenKen involves all four basic mathematical operations AND it is a 4x4 grid!
What do the four corner cells add up to?
Focus on the bottom row. You have a (4) freebie in the third cell. The only solution for the (4+) cage is (3, 1), because a repeated (2, 2) is not allowed. We don’t know the order of (3, 1), but we now know that the bottom row contains (1 or 3, 3 or 1, 4, ?), so the fourth cell must be 3.
Correct Solution: 11
3.2 This 4x4 KenKen is harder than the previous KenKens. Good luck. Don’t give up. There is a hint if you really, really need it.
What do the four corner cells add up to?
Focus on the left column. The only way to fill the (6x) cage is with (3, 2, 1). We don’t know the order, but what does it tell us about the only other cell in that column, the bottom left corner cell?
Correct Solution: 10
(This is your last KenKen... unless you have now caught the KenKen bug? Your teachers might want to introduce KenKen into the classroom, and you can find more for yourself by visiting kenkenpuzzle.com)
KenKen® is a registered trademark of KenKen Puzzle, LLC. Copyright KenKen Puzzle. All rights reserved.
4. Junior Maths Challenge Problem (UKMT)
For parts 4.1, 4.2, 4.3 & 4.4 calculators are not allowed. Remember, the Junior Maths Challenge test does not allow any calculators. However, you can use a calculator for part 4.5. And you can use a calculator to check the answers you have worked out by hand.
4.1 The number 987 654 321 is multiplied by 9.
How many times does the digit 8 occur in the result?
- 1
- 2
- 3
- 4
- 9
- (Not answered)
There does not seem to be any better way to answer this question than to do the multiplication:
From this we see that the digit 8 occurs 9 times in the answer.
4.2 JMC follow-up question What is the surprising answer to 127 × 9,721?
Correct Solution: 1,234,567
4.3 JMC follow-up question What is the surprising answer to 32,800,328 × 271?
Correct Solution: 8,888,888,888
4.4 JMC follow-up question What is the surprising answer to 239 × 4,649?
Correct Solution: 1,111,111
4.5 JMC follow-up question What is the surprising answer to 21,649 × 513,239?
Correct Solution: 11,111,111,111
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.