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Mind Reading

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?

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How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

More Number Pyramids

Age 11 to 14 Challenge Level:

This problem follows on from Number Pyramids.
In the number pyramid below, the number in the bottom left hand corner determines all the other numbers.
If you are viewing the interactivity on a mobile phone, you may need to rotate your screen to landscape orientation.

Try entering some different numbers.
What patterns do you notice?

Here are some questions to consider:

  • If I give you the number in the bottom left hand corner, can you work out the top number without working out the middle layers?
  • Can you explain why some numbers are impossible to get at the top, if you start with a whole number in the bottom left hand corner?
  • If I give you the number at the top, can you find a quick way of working out the number in the bottom left hand corner?

Do your insights still apply if you use negative numbers or decimals?

Can you justify any generalisations that you have reached?
Perhaps you could use algebra to explain your thinking.

Can you adapt what you discovered about 4-layer pyramids to larger pyramids? 

What if the numbers on the bottom layer go up in 2s? Or 3s?

Here is an interactive number pyramid, where the bottom layer can go up by numbers other than one, which can be used to test out your conjectures.