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Top-heavy Pyramids

Age 11 to 14 Challenge Level:

Why do this problem?

This problem is simple to explain yet involves quite a complicated solution process. This problem will hone skills of addition of two digit numbers whilst challenging the organised mathematical thinking of students. The problem may be done by trial and error or with some appeal to algebra.

Possible approach

It is possible for this problem to be done entirely individually but a group discussion may lead to more insights about the strucure of the number pyramid. There are a great number of possible combinations of base numbers; ideally students should be encouraged to understand some of the structure of the pyramid in order to reduce the number of possibilities that they have to try out.

You might initially discuss the problem as a group. Can anyone see any structure or offer a solution strategy? Students could then experiment individually with various combinations of numbers. Encourage students to devise a clear recording system. Encourage them to decide sensibly on the next combination of numbers to try rather than randomly. For example, if a top number is too small then how can the numbers at the bottom be altered to increase this?

You could use a simple spreadsheet to model the pyramid. Could students construct one of these themselves? This is an interesting structural challenge which allows the creation and investigation of larger pyramids.

Key questions

Key questions should lead to understanding the strucutre of the pyramids
  • What is the total for the order $1, 2, 4, 8, 9, 12$? Would we get the same total with a different order? Why?
  • What is the largest possible top number for the pyramid?
  • What is the smallest possible top number total for the pyramid?
  • Which pairs of numbers can be switched without changing the value at the top of the pyramid?

Possible extension

Once an student has found a solution they could be asked these questions:
  • Are there any other possible combinations of $1, 3, 4, 8, 9, 12$ which lead to the answer?
  • What other top numbers are possible? Can you find top numbers which are not possible?
  • Are there any other combinations of 6 base numbers which lead to the top number being $200$?

Possible support

Students who struggle with the level of addition might be provided with a simple spreadsheet to do the calculations. They could also be asked simply to work out $5$ pyramids with different numbers to see who can get the largest number or the answer closest to $200$.