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Marvellous Matrix

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This problem was written for new year $2002$.


Circle any number in the matrix, for example, $608$ as below. Draw a line through all the squares that lie in the same row and column as your selected number.

Circle another number which has not got a line through it, for example, $343$ and again rule out all squares in the same row and column.

Repeat for a third time, then circle the remaining number which has not got a line through it.


Add all the circled numbers together. Note your answer.

Try again with a different starting number. What do you notice?

See if you can work out how this matrix works.
Below is a simpler one which might be easier to investigate.


Can you make a similar matrix which generates a different total?