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Adding All Nine
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
Doodles
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Age 11 to 16
Challenge Level
In Number Sandwiches you may have made sandwiches with the numbers 1 to 3, 1 to 4 and 1 to 7.
Can you use the interactivity below to make sandwiches with the numbers 1 to 5? Or 1 to 6?
Sometimes it is difficult to tell whether a task is impossible, or just very difficult!
Can you convince yourself that it is impossible to make sandwiches with the numbers 1 to 5, and 1 to 6?
Click below to reveal some questions that might help you to explain what is happening:
In a "7-sandwich", how many red squares are covered and how many blue squares are covered?
If it were possible to make a "6-sandwich", how many red squares and how many blue squares would be covered?
Click below to reveal some more questions that might help you develop your thinking further:
If you place a 1 on a blue square, on which colour will you place the other 1?
If you place a 2 on a blue square, on which colour will you place the other 2?
If you place a 3 on a blue square, on which colour will you place the other 3?...
In general, what can you say about the colours on which you place pairs of numbers?
When you try to make a sandwich with the numbers from 1 to 5, or from 1 to 6, what goes wrong?
Which other sandwiches are impossible? How can you be sure?
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.