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Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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Age 11 to 14 Challenge Level:


Joshua Bull  (Brooklands Primary School, Suffolk) explains ...

I did this problem by trial and error. I worked out that D + S = E so neither D or S could be 0.
I chose at random some numbers for E and A and worked out my hundreds column first.


I found these solutions:


0751   7913
+2846   +0246

3597   8159

Are there any more solutions?

Here are some more that have been sent in.....

Alana Asher (Eastbury Farm JMI & Nursery School, Middlesex) discovered the same one as Jason's second solution.

These two came for Alicia Persaud and Priya Gami (Eastbury Farm JMI & Nursery School, Middlesex).

8735   5831
+0612   +0497

9347   6328


Here's another one from Tan Ian Wern (Tao Nan School, Singapore)






Zachary from Clearwater Bay School in Hong Kong has found another different solution:






Laura, Sophia and Sophie from St Michael’s Collegiate School in Hobart, Tasmania, found another different solution:                        



Jayden from Elm Park School in Auckland has found another new solution:


Well done!

Pierre Thomson from Rifton, New York, wrote to tell us that he was working on this problem with his daughter. He managed to write a computer program to find all the solutions and discovered there are 140 altogether! However, some of his solutions (like some of the above) included a zero in the thousands column - this is not how we usually write numbers so you may prefer to ignore these if you are working on this problem.