Math Is Fun Forum
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#1 2014-05-26 22:26:20
Cheeseburgers!
Homer Simpson's bakery sells donuts in 5 packs, 9 packs and 13 packs. Show that it is possible to get any integer quantity of donuts larger than 21.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#2 2014-05-26 22:48:25
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
The Frobenius number of 5 , 19 and 13 is 21.
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.
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#3 2014-05-26 22:50:42
Re: Cheeseburgers!
What?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#4 2014-05-26 22:56:45
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
This problem was worked on a long time ago. The Frobenius number of (a1,a2,a3...ak) is the largest value of n in a1 x1+ a2 x2 + a3 x3 +...+ ak xk = n that does not have a solution in non negative integers. frobenius(5,9,130) = 21 meaning 5 x1 + 9 x2 + 13 x3 =21 has no solutions and anything greater than 21 does.
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.
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#5 2014-05-26 23:03:47
Re: Cheeseburgers!
Thanks for letting me know the bhalo naam
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#6 2014-05-26 23:08:37
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
First understand what I mean:
FrobeniusNumber[{5, 9, 13}]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.
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#7 2014-05-26 23:10:07
Re: Cheeseburgers!
I've already done that ten minutes before...
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#8 2014-05-26 23:17:26
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
Then why do you act as if you have never heard of the term?
The proof is easy.
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.
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#9 2014-05-26 23:29:52
Re: Cheeseburgers!
I looked up the command after you told me what it is called.
Please let me see if i can do the proof.
Last edited by Agnishom (2014-05-26 23:30:35)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#10 2014-05-26 23:32:04
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
The great Sylvester discovered a closed form for frobenius(a,b) but no known closed for is known for any higher ones. But for yours there is.
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.
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#11 2014-05-26 23:33:20
Re: Cheeseburgers!
Because gcd(a,b,c)=1
Yeah?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#12 2014-05-26 23:37:31
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
You are close but you can do better than that.
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.
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#13 2014-05-27 00:46:39
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
What about the proof you said you wanted to do by yourself? Do you have it yet or do you want me to provide 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.
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#14 2014-05-27 01:11:35
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,140
Re: Cheeseburgers!
I've only just looked at this because I don't eat cheeseburgers. 
You should have said it was really donuts! I like them!
You could use an inductive proof. ie Show that if N is possible, then so is N+5.
But what are the starting values?
Clearly 5 pack availability means any number of the form 5n + 0 with n > 0 is OK.
9 pack availability means any number of the form 5n + 4 with n > 0 is OK.
13 pack availability means any number of the form 5n + 3 with n > 1 is OK.
What about numbers of the form 5n + 2 and 5n + 1 ?
Well 13 + 9 = 22 so any number of the form 5n + 2 n > 3 is OK.
And 13 + 13 = 26 so any number of the form 5n + 1 n > 4 is OK.
So the 'seed' numbers for these sequences are 22 = 13 + 9; 23 = 13 + 5 + 5; 24 = 9 + 5 + 5 + 5; 25 = 5 + 5 + 5 + 5 + 5; 26 = 13 + 13.
Thereafter we can generate any n by identifying which class it is in (eg. 5n + 2), selecting the right seed number and just adding 5s.
eg.
47 is in the 5n+ 2 class. So 47 = 13 + 9 + 5 + 5 + 5 + 5 + 5
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#15 2014-05-27 01:23:32
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
Or
You can exploit the properties of the Frobenius number.
Theorem (Lewin 1975):
Plugging in you will get 21 for your problem, which by definition all numbers greater then that are representable.
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.
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#16 2014-05-27 02:49:16
Re: Cheeseburgers!
22 = 13 + 9
23 = 13 + 5 + 5
24 = 9 + 5 + 5 + 5
25 = 5 + 5 + 5 + 5 + 5
26 = 13 + 13
To get a number higher than 26, you just need to add 5's to the extreme right of the sequences...
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#17 2014-05-27 04:29:44
Re: Cheeseburgers!
Oh, its just bob's proof that I discovered
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#18 2014-05-27 15:27:28
Re: Cheeseburgers!
I do not know any formal solution but I did experimentally with a gf:
Finding the taylor series of which gives you:
Looks like everything higher than 21 is possible.
Read on for a proof that all integer quantities > 21 can be expressed as a multiple sum of these quantites:
22 = 13 + 9
23 = 13 + 5 + 5
24 = 9 + 5 + 5 + 5
25 = 5 + 5 + 5 + 5 + 5
26 = 13 + 13
To get a number higher than 26, you just need to add 5's to the extreme right of the sequences...
Analytical Solution:
When the value of the coins are in arithmetic progression, this formula works:
Plugin a=5,d=4,s=2. The answer is then 21
Thanks to bobbym none for the analytical solution.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#19 2014-05-27 15:32:46
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
The first method provides evidence and years ago led me to a proof very similar to Bob's.
The formula is already proved by Lewin 1975 and is ready for use.
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.
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#20 2014-05-27 15:34:08
Re: Cheeseburgers!
Suppose you do not know his theorem?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#21 2014-05-27 15:41:59
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
RIPOSTP. First let me say the author of the problem has made it easy for you. This is typical.
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.
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#22 2014-05-27 15:46:18
Re: Cheeseburgers!
What if an eccentric genii kidnapped me in a prison cell with no wi-fi and said that he won't let me go until I solve it?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#23 2014-05-27 15:59:16
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
You do not understand. The problem has been made to be solved. By giving you the answer (21) you only have to prove that. The teacher or problem poser is not an idiot. He is not going to ask a question that he himself does not know the answer to. How would he check yours if he did not have the solution? This is an important point in problem solving. It was explained to me succinctly by Joe the reppie.
When gAr or EVW post a problem at me, I automatically know there is a solution. Same thing with Calvin. This is much better than the aydin problem. It might have a solution and it might not. It might be a research level problem, I do not know and neither does the poser.
The strangest thing is that problems that people make up, particularly amateurs are almost always of the second type.
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.
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#24 2014-05-27 16:13:59
Re: Cheeseburgers!
But the eccentric genii is not gAr or Calvin.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#25 2014-05-27 21:53:23
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cheeseburgers!
A country has 5 different types of stamps ( 12 , 19, 37, 83 and 117 ). What is the highest denomination that they can not represent using them? Answer without your computer.
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.
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