# Maxi Pyramid

Five integers (whole numbers) are placed on the bottom row of this pyramid. They must satisfy the following conditions:

- No zeros or negative numbers are allowed
- The five numbers on the bottom row must add up to $20$

Each number in the upper rows of the pyramid is formed by combining the two numbers below it, according to the following rules:

- If the two numbers below are even, you add them to get the one above
- If the two numbers below are odd, you take the smaller from the larger to get the one above
- If one number is odd and one is even, you multiply the two numbers to get the one above

Try starting with $4, 6, 1, 7, 2$ on the bottom row. What do you get at the top?

You should have got $60$ at the top. You can see the completed pyramid in the Getting Started section.

**What is the largest top number you can obtain?****Extension:**

What is the largest top number you can obtain if zeroes are allowed?

What is the largest top number you can obtain if negative numbers are allowed?*With thanks to Don Steward, whose ideas formed the basis of this problem.*

The diagram below shows the completed pyramid when the row at the bottom contains $4, 6, 1, 7, 2$.

We received lots of large top numbers. Rachel from Holy Child Killiney in Ireland got 122:

Aila, Lathika, Addisyn and Tamsyn from Frederick Irwin Anglican School in Australia sent in this pyramid with some of their reasoning:

Toby from Pierrepont Gamston Primary School in England wrote:

The most I got was 128 as the top number. On the bottom, it was 4, 3, 4, 5, 4. I did this because I wanted the largest numbers I could shared out and I want to do as much multiplication as possible and as explained, an odd and an even get multiplied.

Alonso and Santiago from Colegio Roosevelt in Peru sent in several Maxi pyramids:

Roisin from Holy Child Killiney sent in this pyramid:

Amélie from St Nicholas CE Primary in England sent in this pyramid:

200

116 84

56 60 24

14 42 18 6

2 7 6 3 2

Olly from Warden Park in England:

Per André from Hakadal Ungdomsskole in Norway sent in this clear explanation and pyramid:

To achieve the highest number on top of the pyramid, according to the given instructions, I had to do the following:

After some time with testing, I decided that the best approach would be to place a high even number (6) in the middle. And then I put a high odd number (5 and 7) as possible on each side, so that I could multiply them with the one on the middle. And then I put 1 on each side, so that it sums up to 20. If I had used a bigger number on each side, the numbers in the middle would have been lower. So the most important thing to solve this, is to put the biggest numbers in the middle.

The advantage by putting the biggest numbers (5, 6, 7) in the middle, is that we also get big numbers (30, 42) in the middle on line 2. If we placed the numbers like this:

We can see that the biggest numbers on line 2 is placed on each side, and not in the middle. This is a disadvantage, because the middle number on line 3 is made up from adding two small numbers.

I.A, Mignione, Saskia.B from Frederick Irwin School managed to get an even larger number at the top of their pyramid:

Ramya, Reef and Caleb from Frederick Irwin School had another idea:

We found out that if you make an odd number on a high level of the pyramid then there will be an even number most likely next to it and they will multiply together to make a huge number.

This was a very good idea, but can you see where they broke one of the rules?

Is it possible to make an odd number on a high level of the pyramid?

Ci Hui Minh Ngoc from Kelvin Grove State College (Brisbane) in Australia thought harder about odd and even numbers:

This strategy helped Ci Hui Minh Ngoc get 222 at the top of the pyramid:

This problem requires students to consider what happens as they try different combination of numbers at the bottom of the pyramid. There are a lot of rules to consider when constructing the pyramid, so students need to check carefully that they have satisfied them all. When they try the extension questions they have an opportunity to practice arithmetic with positive and negative numbers.

The starting row $4,6,1,7,2$ can be used as a joint class activity to introduce the problem. With this starting row the number at the top of the pyramid is $60$ (see Getting Started for all of the resulting numbers in the pyramid).

Students can be challenged to make the biggest top number.

Students could be asked what they notice about all the numbers above the bottom row - is this always true?

*If the numbers on the bottom row are all positive integers, the highest possible top number may be a little under 250.*

**If zeroes are allowed**, the highest possible top number is greater than 250.

**If negative numbers are allowed** it is possible to make huge top numbers...