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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

A grocer has a sack of flour which weights 10 kgs, 2 weights of 200 and 300 grams and a 2-pan balance.

He only sells the flour at multiples of 100 grams, and only if he can measure the desired quantity in only 3 weighings.

What is the probability that he can serve the first customer that enters the shop, who may desire to buy any quantity, starting from 100 grams and up to 10 kgs?

Suppose that each weighing is completed once the 2 pans balance out and also that the bags have negligible weight.

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

OK. By the 1st weighing we can have:

0g

10000g

5000g

100g

9900g

200g

9800g

4900g

5100g

300g

9700g

500g

9500g

Then we must make combinations of the existing balance weights and the ones we have created.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 82,610

Hi anna_gg;

You eliminated all the possible weights that are not multiples of 100. Is that okay?

**In mathematics, you don't understand things. You just get used to them.I have the result, but I do not yet know how to get it.All physicists, and a good many quite respectable mathematicians are contemptuous about proof.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

hi bobbym,

At the monent I'm having trouble at the 3rd weighing because you cannot, for example,

use some already weighed flour (from 2nd weighing )as an extra weight along with another different weight derived from 2nd weighings.

eg. 2100 = 1000 + 500 + 300 + 200 + 100 would be disallowed because it takes 2 weighings to get 1000,

plus 1 weighing to get 500, plus 1 weighing to get 100 and that's 4 weighings.

I think, anna_gg, what is needed here is a notation that includes information about how many weighings an amount took.

Once I've got that it should be a piece of cake. (Made with the flour of course, boom boom! )

Back when I've cracked it.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

LATER EDIT: I am in the process of correcting this 'solution'. I am optimistic that I'll get there some time (... length of time unspecified ...)

I am assuming the grocer has a limitless supply of weightless bags to put his derived flour weights into.

How about subscripts?

A weight that exists at the start will have zero as its subscript.

A weight derived at 1st weighing will have subscript 1.

I did have some others here, not multiples of 100. I have now removed these.

I do not think something like 4850 can later generate a multiple of 100 except by restoring the number that made it.

This conjecture needs proving (or a counter example).

**I believe this is a complete list of 1st weighings.**

These 1st weighing amounts are put into bags and become additional 'weights'.

Now, what can be made at the 2nd weighing?

Theorem: (i) Where different flour bags are used the subscript total to achieve a new weight must be strictly less than the subscript of the new weight. *

(ii) but if the same bag is used in several stages the subscript total to achieve a new weight **need not count this bag repeatedly**

to make this total strictly less than the subscript of the new weight. An example of this is shown later in achieving 1400_3 and 1600_3.

This theorem needs to be modified. My thanks to anonynmstefy for pointing this out.

* so here, 2 > 1 + 0 + 0

Now those 'weights' can also be bagged up to use at the next stage.

3rd weighings

I cannot find a way to do 1400.

But anoninmstefy did:

I think that's it. If you spot any more, please let me know.

Oh darn it! I've just noticed that 5000. That'll be 5000 against 5000.

So the list needs some additions. ..... Tomorrow perhaps. No better still, you've seen the method, I'll leave it as an exercise for the reader.

Additionally, for whatever amount, **x**, that can be measured out, the amount **10000 - x** is also available for sale which doubles the list!

List so far:

I'm assuming the probability is easy to get from here.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi guys

There is the simple and sneaky solution where the grocer just takes meter,measures the 10 kg bag and then just divides it to equal parts and then just draws "circles" around the bag then makes the cuts in the bag,collects the flour that is going to fall out and he takes all the flour that will make the flour in the bag flat,and repeats until he gets enough 100g bags to serve the customer.This is of course a trick solution,and is not the one wanted.

I will start working on the real one.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

I am modifying both my earlier posts to correct my stupid errors.

Sadly this may take some time.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi Bob

I am sure you just missed the solutions.It's not stupid at all.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

hi bob

Found a way for 1400:

*Last edited by anonimnystefy (2012-04-14 10:08:20)*

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

hi Stefy,

You are right! And that means my theorem needs to be modified.

I will add your solution to mine (with due credit of course) so we get a complete list.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi bobbym

Once again I will tell you-I do not need credit!

I just want to make sure in case you haven't seen the edit-Add 1600 and 9400.And don't forget to add 9600!

Here lies the reader who will never open this book. He is forever dead.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi bob

Why have you added 4950 to your list?

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

hi Stefy,

I have been modifying my results to allow for spliting the 10000 either side of the balance using the 200 and/or 300.

So one new possibility is

As I find them, I add them to the list, because I'll forget if I wait until the end.

I think the 1st weighings list is now complete.

The 2nd and 3rd weighings need a number of additions ... frightenly many additions. I think I'll have to let my brain rest a while and come back to the list tomorrow. Do you think anyone will mind if I unhide my answer as it is a real pain to have to deal with it like that.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

I will here make a list of all the quantities that were not mentioned in your earlier post,that I have found:

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

How? My brain just exploded.

Bo b

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi bob

*Last edited by anonimnystefy (2012-04-14 10:51:02)*

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

OK thanks.

This list is going to take a while.

But

He only sells the flour at multiples of 100 grams,

So you were right. I shouldn't have included 4950.

Told you my brain had exploded.

Think I'll call it a day.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

I am trying to figure out if 1900 can be done,but I think that 5050 and 4950 are incorrect. How do you know when you have put 5050 on the left?

Here lies the reader who will never open this book. He is forever dead.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi bob

I am gonna post openly too. Maybe if some of you mods have the time you can hide it.

I will soon or tomorrow post more.

*Last edited by anonimnystefy (2012-04-14 12:24:03)*

Here lies the reader who will never open this book. He is forever dead.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Here's some more:

*Last edited by anonimnystefy (2012-04-14 12:24:46)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

With those numbers we have got all of the possibilities from 100 to 4000,excluding 3900,and by taking the "complements" i.e. the flour that is left in the sack we have also all the possibilities from 6000 to 10000 except 6001. The rest I conjecture to be unobtainable within the given restaints. I will try to prove that in my next post.

*Last edited by anonimnystefy (2012-04-14 12:34:15)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

This is proof that the numbers from 4001 to 5999 are unobtainable within the given restraints:

In every measuring we can obtain only a sack that is the same in weight as the total of the weights of the sacks we had previously. This means that from the 500 total we have at the beginning we can get only a sack weighting 500 grams. So now we have 1000 g. at most. Next we can gain mostly 1000 g. so now have a total of 2000 g. After the third measuring we have 4000 g. maximum. So numbers greater than 4000 cannot be obtained except maybe by obtaining their complement. But wait,if a number is greater than 4000 and less than 6000 then its inverse is in the same boundaries. So it is unobtainable,too! What still needs to be shown is that the numbers in this range cannot be gotten by doing the complement thing somewhere in the process of measuring,and that 3900 and its complement, 6100, cannot be gotten as well.

*Last edited by anonimnystefy (2012-04-14 12:48:33)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Can anyone check if these above are correct?

Here lies the reader who will never open this book. He is forever dead.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,257

I have tidied up post #5 and 'unhidden' it.

I shall stop editing there and rather add new values in new posts as I find them.

Here is a list that, hopefully, contains all values found so far. (apologies if I have missed your values)

I suggest each new contributer copies that list and adds their values to it.

That way we will always have the best list so far at the bottom of the thread.

Stefy: Your posts 20, 21, 22. It looks like you might be going beyond 3 weighings here.

That's why I made the subscript notation; to help keeping track of how many weighings you are using.

My next step will be to add to the 2nd weighings list above, taking account of the 4000 - 6000 new values that I did't spot to start with.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 14,859

Hi bob

I thought you might not be okay with what I did. But look at this:

Let's consider the case for 1900:

We have made sacks of 100,200,300,500 and 900. Then we can just pour the flour from all sacks,except the sack with 100g into a single sack and we have 1900g for the costumer.

Here lies the reader who will never open this book. He is forever dead.

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