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# Factors and Multiples Puzzle

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Xavier Irymple Secondary College in Australia started with square and triangluar numbers. Here is Xavier's work and completed grid:

I decided to work with square numbers on the opposite side of triangular numbers, since the number 1 was relevant to both of them. Then I placed both the triangular numbers and square numbers in their relevant place. Then I just built the rest around those numbers and categories.

Andrea from Norwich School in the UK noticed that some of the categories could not share any numbers. Here is Andrea's method and completed grid, which is different to Xavier's:

I started out by doing trial and error, but soon realised that headings like NUMBERS LESS THAN TWENTY and NUMBERS MORE THAN TWENTY needed to be on the same row. TRIANGULAR NUMBERS and SQUARE NUMBERS both had the number 36, so I had to put them with an intersecting square. I then just filled in the rest of the squares.

Sehar, Saanvi, Dhanvin, Aariz,Vivaan, Sai, Paavani, Utkarsh, Tavish and Dhruv from Ganit Kreeda, Emily, Mason and Summer, Tyler and Frankie and Jakob from Irymple Secondary College, James and Aarush from Norwich School and Dean from Saint Spyridon College in Australia also found categories with little or no overlap. They didn't all choose the same
categories, and all found different grids. Here is Dean's approach, and by clicking below you can see each of the grids.

Dean started this challenge by thinking about the heading cards and where they could be placed based on his knowledge of number properties.

He realised that the 'numbers less than 20' and 'numbers more than 20' needed to be placed on the same side of the gameboard to avoid a clash.

From here, Dean then realised that this mathematical thinking would help him to place the other heading cards around the game board correctly. He

noticed that 'even numbers' and 'odd numbers' also needed to be placed on the same side of the board. Similarly, 'triangular numbers' and 'square numbers' did not have many possible answers on the number cards provided, so Dean realised that it would be best to work with these heading cards and the relevant numbers before attempting the others. In this way, he knew that the remaining number cards
were more versatile and that he could place them in different places because they shared common number properties. With strong perseverance, Dean was able to successfully complete the puzzle independently.

Click to see the grids filled in using similar methods:

Jackson and Declan from Irymple Secondary College and Pranathi and Ananthjith from Ganit Kreeda used systematic listing methods. Here is some of James and Declan's method, and their completed grid (which is different to all the grids we've seen before):

Solving the Factors and Multiples Puzzles was both challenging and rewarding. [We] approached it methodically, starting with a clear understanding of the concepts. First, [we] checked to see what 5 headings fit all 25 numbers then put the 5 headings on one row but if all the numbers didn’t fit with the row and the column [we] would try the strategy again.

These are the Pranathi's lists and grid. Note that, although Pranathi has not ticked 1 as a prime number, it is included as a prime number on Pranathi's grid, which isn't quite right.

Ananthajith's work shows how Ananthajith used listing to create a final grid (which, again, is different to all the grids above):

Isaac and Alexandre used a similar systematic listing method, and found all possible number combinations for a certain arangement of the categories into rows and columns. They used the programming language Python. Click here to open their work.

We received some more complete grids with no methods. These grids are all different to any of the grids above. Click to see the complete grids.