Math Is Fun Forum

atran

Member

Registered: 2013-07-12

Posts: 91

What number is in the middle?

Hi, here follows the translation of a problem:

In a paper, Ida has written on a line 27 positive integers, ordered by size. The first number is 1 and the last one is 25. Ida tells Emil that the sum of all numbers is 127, the sum of the first nine numbers is 21, and the the sum of the last nine numbers is 65. Suffice that information to Emil to decide what number is standing in the middle?

The only thing I could figure out is the sum 41, which is ∑(a[i]) from i=10 to 18. I need to find (if possible) the 14th number.
I need hints rather than solutions, thanks for your help.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What number is in the middle?

Hi;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

atran

Member

Registered: 2013-07-12

Posts: 91

Re: What number is in the middle?

What does "bounding" mean?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What number is in the middle?

In this context, establish upper and lower bounds or intervals that those two numbers can take.

Do you need to see the working?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: What number is in the middle?

Last edited by ElainaVW (2013-12-29 06:20:34)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What number is in the middle?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

atran

Member

Registered: 2013-07-12

Posts: 91

Re: What number is in the middle?

I got it right! I'm re-solving it to post it here.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What number is in the middle?

Very good! Glad you got it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

atran

Member

Registered: 2013-07-12

Posts: 91

Re: What number is in the middle?

OK, here is the solution (this is difficult):

The first number is 1 and the last one 25.
The total sum is 127.
The sum of the first nine numbers is: S[1-9] = 21
The sum of the last nine numbers is: S[19-27] = 65
From the above sums we deduce that S[10-18] = 41

IF S[1-9]=21 THEN 2 < a9
IF S[19-27]=65 THEN 4 < a26
IF S[10-18]=41 THEN 4 < a18

IF S[19-26]=40 THEN 19-26th numbers are equal to 5; this implies a18=5
IF S[10-17]=36 THEN 4 < a17 and therefore a17=5
IF S[10-16]=31 THEN 4 < a16 and therefore a16=5
IF S[10-17]=26 THEN 4 < a15 and therefore a15=5
IF S[10-17]=21 THEN 4 < a14 and therefore a14=5

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: What number is in the middle?

That is basically what I did. Long but easy to follow.

You have:

From equation 3 you can see that a9 is at least 3.

There are 17 variables from a10 to a26.

If a9 was 5 then the variables a10 to a26 would be 85 or more so it is obvious the a9={3,4}.

From equation 4) we have:

or

Obviously a19 = {3,4,5}

Subtracting 3) + 4) from 2) we get:

with the conditions 3<=a9<=4 and 3<=a19<=5.

But we can now see that a19 can not be 3 or 4 so a19 = 5.

We can now also see that a18 = 5. So 5) becomes

Also a17 has to be 5, so we have

Again a16 has got to be 5.

a15 has got to be 5.

Easy to see that a14 can not be less than 5, so a14 is 5.

Since we are working with linear diophantine equations the whole thing can be shortened by using generating functions for parts of the calculations.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.