Missing Multipliers
Problem
The multiplication square below has had all its headings and answers hidden. All of the headings are numbers from 2 to 12.
By clicking on some of the cells to reveal the answers, can you work out what the headings must be?
Once you've worked out what each heading must be, drag the purple numbers to the appropriate spaces. When you think you have cracked it, click "Show the solution" to see if you are right.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
Here are some more Missing Multiplier challenges you might like to try:
If you enjoyed this problem you may also enjoy Gabriel's Problem
Getting Started
Student Solutions
Joey from St Theresa's Catholic College in Australia and Aditya from St Columbas College in the UK sent in completed grids. This is Aditya's work (click here here to see a bigger version):
| Janeen from Westridge School in America only needed to reveal six answers: |
Rhihito from the American School in Japan sent in this result: |
Keshav from Colchester Royal Grammar School and Anh Minh from the British Vietnamese International School, Hanoi in Vietnam described a general method revealing seven squares. This is Keshav's work:
You have to reveal at least seven squares to find all the headings.
Firstly, reveal any square you want. Then, reveal any other square next to it (horizontally or vertically). Using your knowledge of factors, you can then find the heading of that row.
Then, reveal a square next to either of the two that are already revealed, and make sure that square doesn't belong to the heading that you've already figured out.
Now you have two headings. You can use these two headings to find all the others, still using your knowledge of factors, using just the row and column you've formed.
Could Janeen's method be described in this way?
Tuấn Vé from the British Vietnamese International School in Hanoi described a method for larger grids. How could you use what we've seen here to make this method more efficient?
You need to know at least two values of each row to solve the problem, you will need to open it in two vertical lines. So you will need 2$y$ if the table is $y \times y$.
So for 4$\times$4 you need to open 8 values; for 6$\times$6 you need 12;for 8$\times$8 you need 16 and so on.
That's not quite right, since Keshav's method for a 4 by 4 grid uses 7 values, and Janeen's uses 6.
In fact, one in each row and one in each column means $y\times y-1$ because the row and column will cross.
As Janeen noticed, you don't actually need the sqaure in the row and column, because you can find both of those headings from the other rows. So in fact you need to reveal $y\times y-2$ reveals.
Teachers' Resources
Why do this problem?
This problem offers an opportunity for students to consider common factors while gaining fluency in multiplication facts. The interactivity engages students' curiosity and perseverance by challenging them to complete the grid using a minimum number of 'reveals'.
Multiplication tables are often presented with row and column headings filled in, with students challenged to fill in the products. This task inverts that concept, as students can reveal chosen products and work out possibilities for the headings.
Possible approach
If computers or tablets are available, students could work in pairs using the interactivity. Students could try a few examples to get the idea, and then work on the challenge of trying to find the grid headings by revealing as few cells as possible. Once they have developed some strategies, they could try the larger grids that include bigger numbers.
Key questions
Which numbers give lots of possibilities for row and column headings?
Can you find a way to work out the row and column headers using only 6 reveals?
Possible support
Mystery Matrix works in the same way, but some helpful cells have already been revealed.
Possible extension
Gabriel's Problem and Product Sudoku would make nice follow-up activities.