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Domino Numbers

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Domino Numbers


Here is a grid which looks a little like spots on dice or dominoes:
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Can you see why 2 by 2 is 5?
What is 2 by 3? You may like to use the interactivity above to explore.
2 by 4?
2 by 5?
Can you predict what 2 by 10 will be? Explain how you know.

What is 3 by 2?
3 by 3?
3 by 4?
Predict what 3 by 10 will be and explain how you arrived at your solution.

How about 10 by 2?
10 by 3?
10 by 10?
 


 

It is important to encourage pupils to explain why a particular number pattern occurs, not simply spot the pattern. This problem demonstrates the power of visualisation in these processes.

Bernard Bagnall, the author of "Bernard's Bag" on the old NRICH Prime site, told us that he has used this investigation very successfully with many groups of children. He suggests that if pupils record the number of dots in a table (rather like a multiplication square) more patterns can be investigated easily. For example here is a partially completed table which goes up to a 5 by 5 domino number:

  2 3 4 5
2 5 8    
3 8 13    
4 11      
5 14      

Bernard has explored the following ideas from this starting point:
  • Use the table to write down sort of "square numbers", in other words the number of dots in a 2 by 2 domino number, in a 3 by 3, 4 by 4 and so on. Can you see anything special about these numbers?

  • Add the numbers in each diagonal. Can you find a pattern?

  • What happens if you add the four numbers in a 2 by 2 square in the table? How about a 3 by 3 and 4 by 4 etc?

  • What happens when you find the digital roots of the numbers in the above table? (Again, displaying these in a table themselves makes it easier to notice patterns.) Can you see any patterns in the rows and columns?

  • From the table of digital roots, try adding 2 by 2 squares - what do you notice? How about 3 by 3 squares?