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More Number Pyramids

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Well done to those of you who worked systematically with the interactive pyramid on the site. Sian from GSGW school and Scott noticed that the number at the top increases by 8 when you increase the number in the bottom left hand corner by 1. As Scott puts it:

I have found that each time you put a number in, eg. 1, your top number will be 20 but when you put in 2 your answer will be 28 so each time you put in a number one bigger the answer will be 8 bigger.

 

This table of results produced by Esther illustrates this well:

table
 

Niamh from Brighouse High School and Saifuddin started to generalise the pyramid using pronumerals. This was continued by Thomas from New York who generalised the 4-level problem like this:

Well done to Alicia, Laura, Sam, Amber, Chris, Jake, Sam and James from Colonel Frank Seely School, Matt from Newent, Jim and Sadia from Cooper School, Jen, Nikki, Cit, Molly, Ellie, Orla, Georgie, Hayley, Flora and Chloe from NGHS and Esther who also used algebra to describe the pyramid.

 

Tom from Carmel College started off by noticing that:

The numbers in the bottom row increase by 1, the numbers in the 2nd row increase by 2 and the numbers in the 3rd row increase by 4.

 
Then Paul commented that:
All the numbers at the top must be 20 or more and be a multiple of 4.

 

This was continued by Henry from the Latymer School, Karthik, Tom and Mitali who all algebraically showed that the top number is always a multiple of 4. Tom reasoned this by saying:

You only get multiples of 4 at the top since the number at the top in relation to the bottom left number x, is $8x+12$ which can be factorised and written as $4(2x+3)$ proving it is always divisible by 4 (a multiple of 4).
 

Henry from Finton House School and Louisa from New Zealand gave good reasoning on how you can work backwards from the top number without using explicit algebra. Louisa reasoned that:

To quickly calculate the number in the bottom left hand corner when you know the top number you divide the top number by 4 and find two consecutive numbers that sum to this. Then subtract 1 from the smaller of the two numbers and this is the number in the bottom left corner.

 

We had many explanations on how to find the bottom left number from the top number using algebra. James and David from Colonel Frank Seely School gave a good explanation:

Let t = top number and n = bottom left number. Then $ t=8n+12 \Rightarrow n=\frac{t-12}{8} $

Thanks also to Alicia from Colonel Frank Seely School, Ng Xing Yu from Singapore, Karthik from India and Umarah, Meera, Tehillah, Sara, Olga, Daisy, June, Mo and Kassie from NGHS.

 

Mitali continued this by showing that if the number at the top is 48 then the starting number cannot be an integer: $$\begin{align*} 8x+12&=48 \\ 8x&=36 \\ x&=4.5 \end{align*}$$

Nathan from Rushmore Primary explained this by noting:

If we only use whole numbers, the numbers on the top can only be a number in the 4 times tables, but not the 8 times tables. And 48 is a multiple of 8.

 

Sophie from Putney High School began to extend to 6-layer Pyramids. Thomas then said:

The topmost number for five tiers is given by $16x+32$ The topmost number for six tiers is given by $32x+80$.

This was extended by Brittany Howe, Amy Johnson, Livvy Bolton and Abi B from NGHS who gave an example below:

Matt from Newent began to consider an n-level pyramid, and correctly realised that the coefficient of x in the top term, will be $2^{n-1}$. Using Pascal's triangle below, can you work out the rest of the general formula for the top term in an n-level pyramid?