You are Here: Home >< Maths

Combinatorial Question Watch

1. Good afternoon,

I have 6 different non-repeated numbers from a pool of 1 to 59 numbers, let's say numbers n1...n6 for example.

Now knowing that if we add any 2 of the numbers together to equal a 3rd number there are 20 different combinations available.

These are...

n1 + n2 = n3
n1 + n2 = n4
n1 + n2 = n5
n1 + n2 = n6
n1 + n3 = n4
n1 + n3 = n5
n1 + n3 = n6
n1 + n4 = n5
n1 + n4 = n6
n1 + n5 = n6
n2 + n3 = n4
n2 + n3 = n5
n2 + n3 = n6
n2 + n4 = n5
n2 + n4 = n6
n2 + n5 = n6
n3 + n4 = n5
n3 + n4 = n6
n3 + n5 = n6
n4 + n5 = n6

My question is, what is the maximum count of any 2 seperate numbers being added together to make a 3rd number from the 6 numbers, knowing we can use numbers 1 to 59 please.

I hope this makes sense?

2. (Original post by PAB9)
My question is, what is the maximum count of any 2 seperate numbers being added together to make a 3rd number from the 6 numbers, knowing we can use numbers 1 to 59 please.
I think it's clear that an upper bound on the number is 10.

I suspect the final answer is 6, but can't prove it.

An example is: Let the set be {1,2,3,4,5,6}, in order.

Then,
n1+n2=n3
n1+n3=n4
n1+n4=n5
n1+n5=n6
n2+n3=n5
n2+n4=n6

6 in all.
3. Thanks for the reply ghostwalker, it is appreciated.

I was thinking along the lines of 6 but wondered if there was something I was missing considering that the numbers were from 1 to 59.

I applied the same logic to 3 numbers added together to equal a 4th number & 4 numbers added together to equal a 5th number.

I believe for 3 numbers added together there are 4 combinations if the numbers were 1,2,4,7,10,13 for example...

n1+n2+n3=7
n1+n2+n4=10
n1+n2+n5=13
n2+n3+n4=13

...likewise, for 4 numbers added together there are 2 combinations if the numbers were 1,2,3,4,10,16 for example...

n1+n2+n3+n4=10
n1+n2+n3+n5=16

4. (Original post by PAB9)
Thanks for the reply ghostwalker, it is appreciated.

I was thinking along the lines of 6 but wondered if there was something I was missing considering that the numbers were from 1 to 59.

I applied the same logic to 3 numbers added together to equal a 4th number & 4 numbers added together to equal a 5th number.

I believe for 3 numbers added together there are 4 combinations if the numbers were 1,2,4,7,10,13 for example...

n1+n2+n3=7
n1+n2+n4=10
n1+n2+n5=13
n2+n3+n4=13
Also, 1,2,5,8,11,14.

...likewise, for 4 numbers added together there are 2 combinations if the numbers were 1,2,3,4,10,16 for example...

n1+n2+n3+n4=10
n1+n2+n3+n5=16
I think it would be fairly easy to show that with 4 numbers, 2 is the maximum.
5. (Original post by PAB9)
I was thinking along the lines of 6 but wondered if there was something I was missing considering that the numbers were from 1 to 59.
I can see a way of proving that the max number is 6 now for adding just two numbers, based around showing the first two (lowest) cannot be written as a sum, the third and fourth can only be written in one way as a sum of the lower numbers, and the fifth and sixth can only be written in at most two ways. Hence 6.
6. Thanks ghostwalker for your time and input, it is very much appreciated.
Enjoy the rest of your long weekend and thanks again.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: May 2, 2016
Today on TSR

Last-minute PS help

100s of personal statements examples here

More pressure for kinky sex?

Discussions on TSR

• Latest
• See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants

Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.