### Wednesday Puzzle

(Source omitted to discourage Googling; acknowledgements will come next week).

You have a sealed lockbox about a cubic yard in volume, containing \$100,000 in hundred dollar bills. Your balance scale tells you that the box (with the money inside) weighs 100 pounds. You give the box to your friend Al, who flies it to the moon, while you, along with your balance scale, follow in a separate vehicle. Upon arrival, you retrieve the sealed box, put it on the balance scale and verify that it still weighs 100 pounds. You then give the box to your friend Barb, who loads it into her all-terrain vehicle and drives it to your moonbase, with you following along, again in a separate vehicle. When you get to the moonbase, Barb returns your lockbox. You open it and it’s empty.

Who stole your money, Al or Barb?

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#### 34 Responses to “Wednesday Puzzle”

I’m going to go with the following assumptions:

1. The sealed lockbox is airtight
2. On the moon, the lockbox is weighed in a vacuum
3. The 100 lbs. weight you use on the balance scale has a relatively high density (lead?) so that a loss in air pressure does not cause any significant effects through loss of buoyancy

A hundred dollar bill weighs about 1 gram and 1000 bills weight about 2 lbs.
A cubic yard of air at ground level on earth also weighs about 2 lbs. – or to put it another way, an object with a volume of one cubic yard will experience buyoancy of 2 lbs.
Put it into a vacuum, and it will be 2 lbs. heavier.

If the lockbox contained the money when it was weighed on the moon (in a vacuum), it would have been 2 lbs. heavier than the 100 lbs. lead counterweight.
Since it wasn’t, we can conclude that the money was taken by Al.

If the lockbox was not weighed in a vacuum, but rather under normal atmospheric conditions in whatever vehicle I was driving, it was Barb who took the money.

2. 2 2 Jonathan Kariv

It seems almost too obvious that it’s Barb? After all the weight verification with Alf confirms Alf didn’t take anything out. Unless he put something back in of equal weight, and the box is sealed so that sounds ruled out.

Unless I’m missing something which seems likely because steve tends to ask harder/better puzzles than that.

Hedge answer: Google says 1000 \$100 bills weigh around 1kg (2.2lbs). Similiarly the air has some weight (also around 2 pounds). So the box on earth is 96, and the air 2 and the money 2. If Alf manages to take out the money and leave the earth atmosphere air inside then when on the moon the cubic yard of air on the side of scales with the weight (not the box) isn’t there anymore. The atmosphere above the initial yard is the same on both the oon and earth so cancel’s out but the weight of the air on the moon in that yard outside the box is different to the air inside the box.

Actually I think I’ll go with Alf is the thief on grounds of the “hedge answer” and as this would satisfy the good puzzle theorem.

3. 3 3 JohnW

A hundred dollar bill weighs a bit under 1 gram. So \$100K would be a bit under 1kg, call it 2 pounds. Presumably you also have about a cubic yard of air (assuming the volume of the cash is much smaller than the volume of the box), which weighs about the same as the cash, about 2 pounds. So call it a 96lb box with 2 pounds of air and 2 pounds of cash. Assume the seal is airtight.

If Al took the money but left the air, your balance would read about 98lb (a balance scale will read the same value for a given mass in any non-zero gravitational field). If he took the money and the air (i.e., opened it in a vacuum), the balance would read 96lb.

Since the balance read 100lb, we could conclude that Al did not take the money.

Of course, that seems too easy. Maybe you made different assumptions. What if the box initially did not contain any air? Seems like an odd assumption since no mention was made of pumping out the air.

But if the box had no air to begin with, then if Al opened the box in air (1atm of pressure), then he could have replaced the cash with an approximately equal mass of air. But we are told the box is empty at the moonbase. Does that mean empty of air? If you knew the box had no air, and then you opened it in a room with air, you would expect to notice the air rushing in (if you could open it at all with the pressure differential).

Strange puzzle.

4. 4 4 JohnW

I think Advo guessed the answer that the puzzle maker was looking for. But I think it is a poorly written puzzle. If I were using my balance scale on the moon, I would use it in either my or Al’s pressurized vehicle. Why would I want to mess with a balance in vacuum wearing a bulky pressure suit?

5. 5 5 Brandon Berg

Sounds like Al replaced it with an empty box weighing 100 pounds on the Moon, forgetting that a box weighing 100 pounds on Earth should weigh only a fraction of that on the Moon.

But that seems too easy.

6. 6 6 Harold

The density of paper is a bit less than 1000kg.M2. The density of air is about 2kg/m2. We can ignore the air displaced by the paper.

On Earth, the air in the box is balanced by the buoyancy of the box – so we have 98lb box, 2lb money, 2lb air and -2lb boyancy.

On the moon, there is no bouyancy, so if the box had not been opened and still contained the air we would have 98lb box, 2lb money and 2lb air = 102lb. We actually have 100lb, which must be made of 98lb box, 2lb air and no money. Al stole your money whilst it was till on Earth.

7. 7 7 Harold

#5 – I also was thinking the difference between weight and mass was important, but a balance scale was specified as the weighing device. This effectively cancels out acceleration due to gravity since it is the same on both pans, as Advo mentioned. This could be confusing because we refer to other weighing devices as “balances” and “scales”.

8. 8 8 David Pinto

I believe five is right. Pounds is a measure of force, not mass. The box and money should be much lighter on the moon, so Al took the money.

9. 9 9 RPLong

Advo’s answer looks legit, but there are a lot of loose ends. When I fill a box with money, I displace a good amount of air. It seems like a bit too convenient an assumption that there is exactly 2 lbs. of air in a box of unspecified volume. Also, I would never know the weight of a dollar bill without Googling it, and Googling is supposed to be against the rules. (Is this one of those factoids that one learns from doing a lot of logic puzzles?)

Considering the number of assumptions and cheats I have to make in order to reach Advo’s excellent answer, I’m going to say that Barb took the money.

10. 10 10 Jack Snyder

Barb took the money. As Advo, Harold and David Pinto note, the key is that a “balance” scale is used. To elaborate on Harold’s explanation, unlike a pressure scale, which truly measures weight (by gauging how much the gravitational pull of the body on which it sits exerts pressure on the sensors when an object placed on it is pulled downward), a balance scale just uses counterweights to match the mass of the object being measured. Because those counterweights (or “counter-masses,” as they might more properly be called) have the same mass on the moon as on Earth, we should expect that a 100lb box on Earth gets the same reading from a balance scale on the moon, since the balance scale is (presumably) not so sophisticated as to re-label its counterweights when you fly it to the moon. Therefore, the box had the same mass at the end of the flight to the moon that it did on Earth, which suggests that Al did not tamper with it, and proves that it was not empty when Al gave it back. Because the box was empty after Barb had custody of it, that must mean that Barb tampered with the box, which makes her the likely thief. (One could envision a scenario in which Al opened the box en route, replaced the cash with an object of equal mass, and re-sealed it, only to have Barb be foolish enough to pilfer the worthless object that Al placed in the box, but that seems too far-fetched to be a plausible answer.)

11. 11 11 A Leap at the Wheel

“Your balance scale tells you that the box (with the money inside) weighs 100 pounds.”

How, exactly? Do you mean that the scale is balanced with the box on one side, and a pile of weights marked 100 lbs on the other, where the weights exert 100 lbs at earth sea level? Or have I accounted for the fact that the moon’s gravity is 1.6 m/s^2 and not 9.8 m/s^2 and the pile of weights is actually marked 100* 9.8/1.6 or 612.5 lbs, such that they exert 100 lbs on the moon’s surface?

12. 12 12 nobody.really

INFLATION, triggered by Quantitative Easing!

13. 13 13 khodge

#5 – Since it is a balance scale, the relative gravity cannot be the answer since both sides of the scale should be equally affected.

14. 14 14 Harold

#9 ” When I fill a box with money, I displace a good amount of air. It seems like a bit too convenient an assumption that there is exactly 2 lbs. of air in a box of unspecified volume”

Volume was specified as about a cubic yard. I don’t think googling is not allowed, just googling the answer. The 1000 bills has a volume of only about 1L, so not much air displaced.

#8 “pounds is a unit of force” The sooner you guys switch over to SI units the better it will be. Pounds is a unit of both force and mass, which causes even more confusion than is necessary. Usually it is used as a unit of mass – defined as a fraction of a kilogram. There is also the unit pound force, the force exerted by 1lb under normal gravity – Newtons in SI. A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s2. Here a slug is 32.17lbs, not a slimy garden pest and hedgehog delicacy.

15. 15 15 Ken B

My balance scale needs a counterweight. I use as my counterweight and identical box with identical bills in it. Since bills are heavier than air — this is clearer with coins — if my balance still balances Al is in the clear.

16. 16 16 Ken B

Having now read the thread, I like Advo’s #1 and agree it’s the desired solution. Archimedes’s principle applied to air. Note however that my answer in 13 is safe against such trickery.

17. 17 17 cmprostreet

Al took it but both attempted to take it.

Per Advo and others above, the box should have been heavier than the counterweight due to the air in the box (assuming the counterweight did not have air in it, and the moon weighing was done in a vacuum).

If Al had left the box alone it would be heavier than the counterweight. If he took the money the air would have escaped, leaving the box lighter than the countermeasure (assuming Al opened the box in a vacuum). Al was smart enough to replace the weight of the money, but not smart enough to replace the weight of the air.

Later, Barb opened the box to steal the money, but instead found that the box contained not money, but one of her personal effects (placed by Al). She removed it, leaving the box empty upon your arrival.

@RPLong:

I didn’t assume there were two pounds of air in the lockbox. I said that the lockbox (1 cubic yard) was displacing two pounds of air, and thus had 2 pounds buyoancy under normal atmospheric conditions.

19. 19 19 Daniel Hewitt

Box and counterweight would be the same on both earth and moon. Barb took it.

Daniel,
the box displaces 2 lbs of air while the counterweight presumably displaces only a negligible amount of air.
Consequently, the box will be heavier if weighed under vacuum conditions as the box loses 2 lbs of buyoancy as soon as you put it into a vacuum.

To several others: It’s not about how much air the box contains, it’s about how much it displaces.

21. 21 21 RPLong

Thanks for the correction. I also see that “1 cubic yard” is contained in the body of the original post, which I had missed before.

Looks like you nailed it.

22. 22 22 Jeremy N

I mostly agree with Advo. It is most likely that Al took the money. However, it is possible he did not. If Al arrived with sufficient time to open the box on the moon before my arrival (and the box contained the volume of air before being opened), he could have just looked at the money and allowed the air to escape. The box would then weigh 100 lbs. on the moon with the money.

23. 23 23 Ken B

For those having trouble seeing Advo’s point, imagine the balance uses lead weights, and now try the weighing under water. Under water you can see the box will press less on the scale due to bouyancy — which is caused by displacing more water than the lead weight. This is Archimedes’s “Eureka”. The same thing applies, but with the bouyancy effect reversed, in the air on earth vs no air on the moon.

24. 24 24 Daniel Hewitt

Thank you, that makes sense.

25. 25 25 Michael Stack

Since it is a balance scale both the weight and the counter-weight would be equally affected by the change in gravity, so it would still measure 100 lbs on the moon. Therefore I believe Barb took it.

26. 26 26 Michael Stack

27. 27 27 Ken B

A random act of music, rather a bouyant one. http://www.youtube.com/watch?v=MnK__qtMfRc
Steve Martland died last year.

28. 28 28 Floccina

Do the levers in the balance scale need to be adjusted for he moons gravity?

29. 29 29 David Friedman

It isn’t clear what your “still weighs 100 pounds” means. Since you specified a balance scale, I’m guessing that you mean it balances with the same weights as before. But the weights weigh much less now–weight isn’t mass–since you are on the moon. So balancing with the same weights means it weighs much less, so it doesn’t still weigh 100 pounds.

Assuming my guess is correct, if your weights are denser than the box with the money in it, as seems likely, weighing in the moon’s vacuum should show the box heavier than before, since the measured weight is no longer being reduced by the buoyancy due to the earth’s atmosphere. Since it doesn’t, at least some of the money has been removed. To figure out whether all of it was I would need to know the volume of the weights you are using in order to calculate whether the change in apparent weight that should have been observed due to eliminating buoyancy is equal to the weight of a thousand hundred dollar bills.

I answered this after reading the first few words of the first comment here, but nothing else–I’m guessing that I am not the first person to solve the puzzle.

30. 30 30 Mike H

* is the moon weighing done in a vacuum?
* if so, is the box airtight?

If the moon weighing is done under normal atmospheric pressure, it doesn’t matter of the box is airtight or not. The fact that it weighs in at 100lb shows that the contents have the same mass. Barb is guilty. Or they’re colluding – Alf stole the money, replaced it with moon rocks, then Barb took the moon rocks out. When you search her vehicle, she just says “I ain’t got nothing but these moon rocks” But I digress….

Now, let’s suppose the moon weighing is done in a vacuum. If the box airtight, the 2 lb of air in the box is no longer buoyed up by the earth’s atmosphere. If the money is still inside, the box should clock in at 102 lb. Alf is guilty if the box is airtight.

On the other hand, if the box is not airtight, the loss of the 2lb of buoyancy is balanced out by the loss of 2lb of air. Barb is guilty if the box leaks.

31. 31 31 Neil

If the box is filled with bills, I don’t think there will be much air to worry about. You can’t have a cubic yard of money and a cubic yard of air occupying the same space.

32. 32 32 dave

the box isn’t filled with bills. there are 10 100 bill bundles of 100 dollar bills. the bills take up about half a shoebox worth of space.

33. 33 33 Pete

A balance scale compares the weight of the object being measured to the weight of the hunks of metal on the bars (or the hunks of metal you put on the other side of the scale). Everything’s weight will change proportionally, so the hundred pounds should still read one hundred pounds on the moon, since both the safe and the hunks of metal are 1/6 their earthly weight.

I guess that means it was Barb.

34. 34 34 Jack Coupal

Al stole it. It has to do with weight on earth as against weight on the moon.