Conundrums

[Martin]

Try this one....

The Bank of Puzzlania has uncovered a plot to swap the gold in their vaults with counterfeits. It was determined that all the gold bars in three of the Bank's seven vaults were replaced with counterfeits. The other four vaults were uncompromised.

The plot was foiled through the poor math skills of the thieves: while the real gold bars weigh ten kilograms, the counterfeits all weighed nine kilograms.

You've been tasked with determining which vaults have real gold, and which are just gold-plated bars of platinum. (The thieves were also not financial wizards.)

The Bank Director has made the following generous offer to you: If you can determine the counterfeits using just one weighing on a scale, you can keep one bar as a souvenir.

Here are the rules:
This is a scale, not a balance, but you can weigh as many bars together as you like.
Only one weighing.
The bars will be handled by professional guards, so you won't have a chance to "feel" their weights.
Each vault contains several hundred bars.
The guards have requested that you try to keep the number of bars you need to a minimum.


I will give you the solution in a couple of days time......but a virtual lollipop goes to the first person to come up with a solution........

It's not a particularly easy one, so if there are no takers then I'll give you an easier one.

Just how clever are the people who visit this MB, I wonder???

 

[A Sheep]

There are many possible answers to this query.

First, just weigh one bar from each of the three vaults that have been broken into. For instance, if they have a hole blown into the wall or the door has been rendered unlockable.

Second, Platinum is worth more than 10 times as much as Gold, so the bank should just be happy with their new found fortune.

Third, Gold has a melting point of 1064.43 degrees' Celcius, while Platinum has a melting point of 1772 degrees' Celcius. Ipso facto you could easily just place seven of the bars on a table, turn on a torch to 1100 to 1600 degrees' Celcius and melt the Gold right off of the Platinum.

Forth, if the theives had just Gold-plated some Platinum, the Gold-plated bars would not have the purity percentage number etched into them; as do all Gold bars produced by mints.

Fifth, and the truly easiest option is, due to the fact that Gold weighs 19,300 kg per cubic metre and Platinum weighs 21,450 kg per cubic metre, the Platinum bars that are plated with Gold would be smaller. So no weighing is neccessary, just look at the size of the bars, the three vaults with the smaller bars are the vaults filled with the "counterfiet" Platinum bars.

 

[Martin]

(lol) Nice answers.....

 

[lucypevensie]

I showed this to my husband and he came up with something. I'll have to sit down with him and make sure I understand it before I post his answer. I'll get back to you.

 

[Martin]

It was a very hard one......but if you don't want to know the solution, pass over the rest of this post:





SOLUTION (long):
You’ll need a total of 51 “gold” bars taken from the seven vaults as follows:
1 bar from the first vault,
2 bars from the second vault,
4 bars from the third vault,
7 bars from fourth vault,
13 bars from the fifth vault,
24 bars from the sixth vault,
and 0 bars from seventh vault.

51 gold bars should weight 510 kilograms (ouch!). Place them all on the scale and see how much they actually weigh. But they’ll inevitably weigh less than that, because three vaults contain counterfeits.
Depending on which vaults contained counterfeits, you’ll see different deficits. For example, the heaviest the scale could read would be 507. And if it did, it’s light 3 pounds. The only combination that could yield that amount is 0 + 1 + 2, or the first, second and seventh vaults.
Below are all the possible vault permutations, proving that each combination is unique:
3 = 0 + 1 + 2
5 = 0 + 1 + 4
6 = 0 + 2 + 4
7 = 1 + 2 + 4
8 = 0 + 1 + 7
9 = 0 + 2 + 7
10 = 1 + 2 + 7
11 = 0 + 4 + 7
12 = 1 + 4 + 7
13 = 2 + 4 + 7
14 = 0 + 1 + 13
15 = 0 + 2 + 13
16 = 1 + 2 + 13
17 = 0 + 4 + 13
18 = 1 + 4 + 13
19 = 2 + 4 + 13
20 = 0 + 7 + 13
21 = 1 + 7 + 13
22 = 2 + 7 + 13
24 = 4 + 7 + 13
25 = 0 + 1 + 24
26 = 0 + 2 + 24
27 = 1 + 2 + 24
28 = 0 + 4 + 24
29 = 1 + 4 + 24
30 = 2 + 4 + 24
31 = 0 + 7 + 24
32 = 1 + 7 + 24
33 = 2 + 7 + 24
35 = 4 + 7 + 24
37 = 0 + 13 + 24
38 = 1 + 13 + 24
39 = 2 + 13 + 24
41 = 4 + 13 + 24
44 = 7 + 13 + 24

So now you have a lovely counterfeit bar as a souvenier. Which, as of today’s prices, would be worth about $18,400 US. Not bad for a days work.

 

[Martin]

Here's an easier one, which may you well have come across before........ (let's face it, no conundrums are easy otherwise they wouldn't be conundrums:

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person.
Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar?

 

[lucypevensie]

Here's what we came up with for the bank vault one.

1 bar from vault 1, 2 bars from vault 2, etc, all the way up to 7 bars from vault 7. Their weight would equal 280 kilo IF they were all gold , but they're not all gold. So there will be a difference in weight. Whatever the difference is determines which vaults are the fakes. The smallest possible difference is 6 (vaults 1,2,3) and the highest possible difference is 18 (vaults 5,6,7) Say your total weight is 270 kilo. The difference is 10. Only one combination of three numbers in the 1-7 range equals 10 (2,3,5)

Does this make sense?

Martin, is there any flaw that you can see in this solution?

 

[Patty]

Originally posted by Martin
Here's an easier one, which may you well have come across before........ (let's face it, no conundrums are easy otherwise they wouldn't be conundrums:

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person.
Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar?

Hey Martin,

The three people actually ended up paying ~$9.67 each, because the bellboy's $2.00 came from them. The three people paid $9.00 for the room and 2/3 of $2.00 for what the bellboy took.


Patty

 

[pgp_protector]

Back from the Dead


Any more ?