What do you think?

Fruityloop · 2010-05-30 16:37:54

A bunch of quarters, dimes , nickels and pennies has an average value of 14 cents. If a quarter is replaced by 25 pennies the average would drop to 7 cents. Or if a nickel was replaced by 5 pennies the average becomes 12 cents. What is the largest amount of nickels you can have?

Thank you for the interesting and challenging problems Bobbym.

Last edited by Fruityloop (2010-05-30 16:42:24)

bobbym · 2010-05-30 17:01:13

Hi Fruityloop;

Haven't seen you in a while. Thanks for looking at the problem. That is correct! Very good! How did you go about getting your answer?

phrontister · 2010-05-30 20:03:23

HURRAH + HURRAY = PUZZLES

Last edited by phrontister (2010-05-30 20:48:29)

bobbym · 2010-05-30 20:37:54

Okay, checked. Your answer is correct. That is a solution too. Please hide it and good work!!!!

bobbym · 2010-05-31 00:03:56

Try this one:

POTATO + TOMATO = PUMPKIN

Fruityloop · 2010-05-31 00:16:31

How did you go about getting your answer?

Well I got two equations...

Then it was just trial and error.
Yeah, I haven't posted in a while, I'm going to try to become more active on the forum.

Last edited by Fruityloop (2010-05-31 00:18:56)

bobbym · 2010-05-31 00:33:48

Hi Fruityloop;

Your answer suggests a generating function approach on 24x + 9y + 4z = 312. But I haven't been able to go any further with it then trial and error,

bobbym · 2010-05-31 01:00:53

Hi ;

These are my equations:

Solve the underdetermined simultaneous set of equations:

from those 2 parametric equations we can solve for n:

By varying d and q we can try to maximize n. Better seems to just vary d and q until 12 is reached for n.

phrontister · 2010-05-31 04:55:47

Hi Bobby,

POTATO + TOMATO = PUMPKIN

Last edited by phrontister (2010-05-31 17:51:44)

bobbym · 2010-05-31 09:18:10

Hi phrontister;

That is correct. Well done. Do you think you can come up with a solution method using 7? A method that LB couldn't do?

phrontister · 2010-05-31 13:22:01

Now there's a challenge! I don't think that way, so it'll take me a while to suss that one out. I'll give it a go, but don't wait by the computer for my answer!

bobbym · 2010-05-31 14:32:01

I am waiting. Of course I am waiting. Why would I not be waiting. Waiting is what I do. What I live for. I was born to wait. I always wanted to live in Kuwait, Wanted to be a waiter all my life. Guess what I am doing now.

bobbym · 2010-06-04 11:13:21

How many consecutive integers can you find from 0 to n that sum to 123456?

A says) Jeez you need a computer!
B says) No you don't, there arent' any and I can prove it.
C says) Prove it!
D says) There are 3 solutions.
E says) What are consecutive digits and why should we sum them?

Who is right, please hide and back up your answer.

phrontister · 2010-06-05 01:15:00

Hi Bobby,

to "Consecutive integers that sum to 123456".

bobbym · 2010-06-05 01:23:01

Hi phrontister;

Correct! That's all I found too.

bobbym · 2010-06-05 13:26:57

How many consecutive integers can you find from 0 to n that sum to 1 000 000 000?

A says) You will definitely need a computer now.
B says) No you don't. It is easier than the last one, if you just resist those God awful things.
C says) I want my Mommy!
D says) There are 3 solutions. There are always 3 solutions. The right one, the wrong one and my solution.
E says) Cogito ergo sum. Which means add them up by hand.
F says) Hey E, that's not what that means.

How many are right!

Please hide and back up your answer.

phrontister · 2010-06-06 03:31:28

Hi Bobby,

Integers that sum to 1 000 000 000:

bobbym · 2010-06-06 03:56:20

Hi phrontister;

Excellent work! How long does it take by the method you are using?

phrontister · 2010-06-06 05:09:12

About a minute per solution to enter the figures into my calculator and write the results onto paper...plus the overall time it took me to think about how to do it (probably a couple of hours, not helped by having one eye on the problem and the other on watching Nadal's demolition of Soderling).

With this method there wasn't any time wastage on non-solutions, and I only entered those into my calculator which I 'knew' would work. I haven't actually proven that these are the only solutions, but from the pattern that I saw in the first puzzle I hoped they were, and simply followed that pattern.

After finishing the solutions I quickly checked their integer sums in Excel with an adapted version of the formula n(n+1)/2.

Last edited by phrontister (2010-06-06 06:49:33)

bobbym · 2010-06-06 07:31:29

Hello;

I have a solution that I have adopted for 7. It is not my solution, I believe it is an olympiad or putnam problem solution. It is instantaneous. They say it can be done by hand, but I think not.

phrontister · 2010-06-07 01:15:02

Hi Bobby,

My method's quite easy to do by hand (once you get used to it), but I've adapted it for Excel.

I've tried it out on about ten options, and they all work. I used LB to brute force a check on the number and content of solutions for each problem.

I won't bother with trying to get Excel to automate the total solution. It probably can be done - maybe through VBA by somehow finding all the combinations of the elements of a prime factorization - but that's beyond me.

Here are images of my spreadsheets of the Excel solutions for 1000M and 141723.

It's nice to see the spreadsheet cells update correctly to the new info when inputting a new problem.

Last edited by phrontister (2010-06-07 02:00:54)

bobbym · 2010-06-07 01:53:16

Hi phrontister;

Looks good to me. If you will give me a little time I am cleaning up and formatting my approach. Will take some time.

bobbym · 2010-06-07 02:21:13

Hi phrontister:

This is the sum you are working with.

FactorInteger[2*1000000000]

foo[l_]:=Times@@l;

The next line needs adjustment for each new problem:

two=Union[foo/@Permutations[{1,2,4,8,16,32,64,128,256,512,1024,5,25,125,625,3125,5^6,5^7,5^8,5^9},{2}]];

ans=Solve[{(b+1)==f,(b+2 m)==2*1000000000/f},{b,m}]/.f->two;

Select[MapThread[List,{ans[[1,2]]//Last,ans[[1,1]]//Last}],IntegerQ[First[#]]==True && First[#]>=0&]

This may seem a little intimidating but it has 2 advantages. No loops, instantaneous answers and mathematical certainty that you got them all.

phrontister · 2010-06-07 03:28:07

Thanks for that, Bobby...I'll try to understand what's happening there. It's gonna send me to the help files! The feature I like best is that it catches all the answers for sure. But how one can be certain of that I wouldn't know.

Last edited by phrontister (2010-06-07 03:32:32)

bobbym · 2010-06-07 03:33:04

Hi phrontister;

Run it and see the speed, also remove the semicolons at the end of each line to see what it outputs.

It is currently set up for any power of 10. For instance 1000,10000,1000000,100000000000000000000...

Math Is Fun Forum

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