Pyramid Investigations 1 – Pyramid Number Patterns

Algebra Level 1

How many circles are in the final group?

This problem is part of the Pyramid Investigations Set .

16 9 18 15

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Arron Kau by Arron Kau

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30 solutions

Prasun Biswas
Mar 29, 2014

If you observe the diagrams, then you can see that the total number of balls on each diagram is equal to the square of the no. of rows in each diagram. That means if there are n n rows, then the no. of balls = n 2 =\large n^2 .

Following this pattern, we can see that the last diagram has 4 4 rows, thus the final diagram has balls = 4 2 = 16 =4^2=\boxed{16}

180 Helpful 0 Interesting 0 Brilliant 0 Confused

just count the total number of balls

HEt Patel - 7 years, 2 months ago

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I also just counted the balls

Kuttiyam Srinivasan - 7 years, 2 months ago

Me too.

Fuad Hasan Ifty - 7 years, 1 month ago

I did the same

Jubin Choudhary - 5 years, 6 months ago

That's what I did

Jeco Palay - 7 years, 2 months ago

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To verify the formula :)

Kuhu Kinnori - 7 years, 1 month ago

I thought it was 1+3=4; 4+5=9 and so add the next odd number 7 to 9 and get 16.

Oliver Daniel - 7 years, 2 months ago

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i too thought the same way

faizan ansari - 7 years, 1 month ago

What I thought too

Grant Norsworthy - 7 years, 1 month ago

That's what I did and then counted just to make sure.

Ryan Trebilcock - 5 years, 9 months ago

The set of odd numbers is also the set of the differences between square numbers so you get the same answer of you think of it like that

Samuel Yaroson - 5 years, 5 months ago

It's basically sum of arithmetic progression

Aarambh Sanjay - 7 years, 2 months ago

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Yes, and sum of first n n odd natural nos. is calculated by the formula --> S = n 2 \boxed{S=n^2}

Prasun Biswas - 7 years, 2 months ago

I thought it was 1+3=4; 4+5=9 and so add the next odd number 7 to 9 and get 16.

Hanaa Ahmed - 7 years, 1 month ago

im just 9 just do 9+7

Bill Nguyen - 7 years ago

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That is not the correct formula

Erin Molton - 5 years, 5 months ago

its 16. just count the balls

Moazam Tariq - 7 years, 1 month ago

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haha me too...

Sruthi Lakshmi - 5 years, 11 months ago

Good to know the math behind the problem!

I realized this retroactively but a visual representation for that formula would be to take the rest of the balls after the peak of any triangle and fit them into a square. It is literally n squared.

Ilana Antonio - 5 years, 10 months ago

Thanks a lot Prasun. Could it be that the area of a triangle is the square of its height? :-O

Mags Gertos - 7 years, 2 months ago

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Actually, the basic formula for finding the area of a triangle with base b b and height h h is A = 1 2 × b × h A=\frac{1}{2}\times b\times h . Now, if b = 2 h b=2h , then the area will be calculated by h 2 h^2 . If you have any more doubts, reply me in the comments.. :)

Prasun Biswas - 7 years, 2 months ago

How can be that? I am just a novice, wanting to learn Sir.

Jeco Palay - 7 years, 2 months ago

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Which part of the solution are you not able to understand ?? Feel free to reply in the comments. I would be glad to clarify any doubts you have ! :)

Prasun Biswas - 7 years, 2 months ago

this is a series of" n" no. of rows then square of "n"

Rajkumar Chauhan - 7 years, 1 month ago

I did this as well

Juan Pablo Ferres De Souza Lemos Escobar - 5 years, 10 months ago

Não havia pensado dessa maneira, mas observando os diagramas e fazendo uma projeção faz sentido sim. Didn't thought this way, but that's correct too, it makes all sense when you observe the diagrams and make a projection...

Luis Eduardo Carzino Beira - 5 years, 9 months ago

lol everyone just counted it😂

Matt Duffy - 5 years, 6 months ago

the pattern is sq. of the fig no. i.e 4*4=16

Mayank Chaurasia - 7 years, 1 month ago
Murilo Honório
Apr 20, 2014

1²=1, 2²=4, 3²=9, 4²=16

3 Helpful 0 Interesting 0 Brilliant 0 Confused

1+3=4; 4+5=9 and so add the next odd number 7 to 9 and get 16.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Abdul Gani
Dec 20, 2015

Count the balls

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Samih El Bouzidi
Apr 27, 2014

1^2
2^2
3^2
4^2=16


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Earvinj Alancado
Apr 17, 2014

It's an application of exponential function, the no. of rows represent as "n" and the no. of balls represent as "n square"

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Debasis Rath
Jan 6, 2016

Even by seeing the diagram see the length of side to height ratio is equal I mean like3:3 BALL RATIO IN THIRD

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Shakaib Haider
Dec 6, 2015

By counting the number of balls, or by the square of no. of rows.

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Dom Jones
Nov 10, 2015

I refused to count the balls, but the pattern is +3,+5,+7 an so on increasing each by 2

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Chad Johnston
Oct 30, 2015

I did the area

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EngrTahir Ronaldo
Oct 25, 2015

simply count the balls

0 Helpful 0 Interesting 0 Brilliant 0 Confused

This problem is a good example to show that (n+1)^2=n^2+n*2+1

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Ellis Rennie
Oct 14, 2015

Just count them...

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Sai Sandesh
Oct 12, 2015

In the 1st set 1^2=1, in 2nd case 2^2=4, in 4th case 4^2=16

0 Helpful 0 Interesting 0 Brilliant 0 Confused

boring... too easy.

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Brendix Emata
Sep 6, 2015

It is actually a sequence of the squares of natural numbers: 1^2 , 2^2 , 3^2. So, the next is 4^2 which is 16. Simply easy.

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Matthew Hayes
Aug 31, 2015

the number of balls in each row goes up by 2 each time: (1) 1 (2) 1+3=4 (3)1+3+5=9 therefore, the next row must have seven balls: (?)1+3+5+7=16

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Kathy Man
Aug 28, 2015

The pattern is based on numbers that are perfect squares.

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Ishita .S
Aug 25, 2015

If you carefully observe the diagrams, then you can see that the total no. of balls on each diagram is equal to the square of the no. of rows in each diagram. That means if there are rows, then the no. of balls . Following this pattern, we can see that the last diagram has rows, thus the final diagram has balls . ( you can also count and find the answer)

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Abdul Moeed
Aug 21, 2015

1 +3 +5 +7 first it is 1 then 1+3 then 1+3+5 and then finally its 1+3+5+9=16 so its like n^2 where n is the numer of term in odd series like 3 is second term in odd series so 2^2=4

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Senthamil Arasi
Aug 20, 2015

first pyramid 1^21=1 second pyramid 2^2=4 third pyramid 3^2=9 forth pyramid 4^2=16

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Abbas Majeed
Aug 10, 2015

see the height of triangle and take square of it . i.e. height=4 ...... it means that circles=4^2=16

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1+3 =4; 4+5 = 9; 9 + 7 = 16

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Adarsh Mahor
Jul 16, 2015

Don't calculate the balls it consumes time See all are square 1. 4, 9, 16

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I thought the same way.

Kathy Man - 5 years, 9 months ago
Sam Javier
Jul 12, 2015

I don't know but the answer is too obvious: just count the balls. Though the point here is to know the Mathematics behind every problem.

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Zohair Khan
Jul 9, 2015

if you observe, you would find a certain number pattern.

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Chloe Michelle
Jul 5, 2015

Just Counted .

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Dustin Trinidad
Jun 17, 2015

if you know how to count to 16 you're fine.....

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Muhammad Imanata
Jun 15, 2015

Fyi,there is formula a+(n-1)b+c/2(n-1)(n-2)

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Susobhan Patra
Jun 14, 2015

in place of giving balls on the final figure, if u can just put the number of rows of the balls and ask for total number of balls, then it will be quite though provoking. :D good luck

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