# More Puzzles

More Puzzles May 30, 2018

I’ve posted puzzles before (part 1, part 2) because of what they can teach us about how the human brain works. Spoiler: it works imperfectly.

You probably remember this puzzle:

A bat and a ball cost \$1.10 in total. The bat costs \$1.00 more than the ball. How much does the ball cost?

An answer will probably pop to mind—10 cents—but that’s wrong. (This and related puzzles were discussed in part 1.) Whether you give the intuitive answer or analyze further to find the correct answer says something about how you think. This is explored in an article provocatively titled, “3-Question Quiz Predicts Whether You Believe in God.”

Here are some more puzzles, just for fun (hints and answers are at the bottom).

#### Quick puzzles

1. “Jack is looking at Anne, and Anne is looking at George; Jack is married, George is not. Is a married person looking at an unmarried person?” Answers: (a) Yes, (b) No, (c) Not enough information. Source

2. Start with 100, then add 10% and then subtract 10%. How much do you have now?

3. Cup 1 holds milk, and cup 2 has an equal amount of coffee. Take a spoonful of milk from cup 1 and pour it into cup 2. Now take a spoonful from cup 2 and pour it into cup 1. Which cup is now more concentrated? Answers: (a) Cup 1 has a higher fraction of milk than cup 2 has coffee, (b) cup 2 has a higher fraction of coffee than cup 1 has milk, (c) the fractions are equal. Source

#### Puzzles about the real world

4. What happens to a helium balloon in a car when the car accelerates? Source

5. Why is a mirror left-right reversed? Why isn’t it top-down reversed, too?

6. Your canoe overturns, and you swim to shore. You’re looking for shelter, and it’s getting dark. Luckily, you find a cabin. Inside, there’s a box of matches and a kerosene lamp. While there is kerosene in the base of the lamp, the wick is too short to reach it. You’ve checked the cabin, and there’s no more kerosene or wicks. How do you light the lamp to signal for help?

#### Math puzzles

7. “Three people with different salaries need to find out their average salary without revealing individual salaries to each other. How?” Source

8. Three people rent a hotel room for \$60, so they each pay \$20. Later, the manager realized he charged too much—it was only supposed to be \$55. He sends a porter to the room with \$5. Each of the people keeps \$1, and they give the porter the remaining \$2 for a tip. So now each person has paid \$19 for the room (\$19×3 = \$57) and the porter got \$2. But they started with \$60, and \$57 + \$2 = \$59! Where is the other dollar? [I saw this one about 50 years ago.]

9. Joanne and her friends are seated around a large table. A dish with 25 cookies is pass around. Each person in turn takes a cookie and passes the dish along until it’s empty. The dish might go around once or several times, but the only rule is that the dish must start and end with Joanne (that is, she takes the first and last cookies). What are the possible values for the number of people sitting at the table? (h/t commenter Dave Gardner)

Whatever Nature has in store for mankind,
unpleasant as it may be,
men must accept,
for ignorance is never better than knowledge.
— Enrico Fermi

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• Herald Newman

I’m not sure that I agree with your answer to question 9. If there were 25 people in total at the table (including Joanne) then Joanne would take the first cookie, the other 24 people would each take a cookie, and there would be none left for Joanne to finish. So I’m not sure how 25 people could be a correct answer. Am I misunderstanding something about this?

I think the valid answers are 1, 2, 3, 4, 6, 8, 12, and 24, people in total.

• MR

I’m confused. Where is the answer for 9? I’d agree with your comment on 25, but 1 and 2 would be ruled out as well, no, since Joanne has friends, plural, which would mean at least 3?

• Herald Newman

The answers can be found by clicking on the “continue reading” button at the bottom of the article, or following [THIS LINK].

• MR

Totally missed that, thanks!

• Kev Green

Yeah, I’m pretty sure that you’re right (except for the point about 1 and 2). In the equation nm +1 = 25 n has to represent the total number of people, not just the number of friends. Otherwise you have Joanne getting exactly one cookie instead of one more than everyone else.

• Hmm. Yes, I may be off by one there. Good catch.

• Otto

Dammit Bob! My head was already spinning from some of the others…Not helping…lol

• Lucy

I would say the answer to Problem 1 is “not enough information” for the following reason: We don’t actually know what species or sort of being two of the individuals are. We know Jack is human for sure, because only humans can get married, but the other two individuals – well, one of them definitely isn’t married, and the other, well, we don’t know if she is. Which means that George, at any rate, might be a dog (or even a monkey – we’ve all heard of Curious George). If Anne is looking at a dog, monkey, or some other nonhuman animal (assuming Anne is human) well, then the answer would be no. But we are not told the species of the individuals whose marital status is not mentioned, so no, not enough information.

Speaking of species, cabins, log or otherwise, are generally found in environments with a lot of trees, which means there is fallen wood, which means you can build a fire. So the answer to problem 6 is “go out and gather wood (kindling, tinder, and bigger logs) and build a fire, using a combination of kindling and kerosene to start it”. And this is a remote area, which means nobody mysteriously cleared the trees, and also deserts don’t mysteriously form around lakes (nor is it a prairie, there is very little prairie left, carefully preserved as it is, and people don’t take their canoes there anyway – the only problem was someone was incompetent with the kerosene wick, that’s all). So there’s definitely wood to build a fire – if there wasn’t, it would be because there are other people there and you can get help anyway. And besides, fires make more light than a kerosene lamp, whereas if you use water to raise the level of the kerosene, you might wreck the kerosene and make the lamp sputter prematurely, wasting kerosene in the process. And if the wood is wet, you hole up for shelter in the cabin, take the wood into the cabin with you, and wait for the wood to dry, because water evaporates. Then the next night you build the fire. That is, if you lost your canoe, or the oar to steer it with – the puzzle doesn’t say if you did. Because you see, when canoes overturn, they can be flipped back right side up by one who can swim. So if you still have your canoe and your oar, the answer is “flip your canoe right side up again – swim to shore with the canoe and oar in tow if you have to, but flip it back upright. Then shelter in the cabin for the night, then when you wake up again, canoe back to the other side of the lake”.

• Kevin K

You could also just put rocks in the lamp so that it raises the level of the kerosene — dry rocks, of course.

Or, you could tear a strip of cloth from your clothing (cotton undergarment would probably be best) and use it to extend the wick.

I like your first answer, though. Build a fire with the matches and the kerosene as a starter. That way, you can signal and keep warm. Heck, with good kindling, you probably don’t even need the kerosene.

• Greg G.

I think you could put water in the lamp. The kerosene would float.

• Kevin K

I thought that might be the case, but wasn’t sure. Rocks would be a sure thing.

• Greg G.

That would leave a lot of kerosene in the bottom of the lantern.

• Kevin K

Perhaps…but I just go back to my days learning how to fly. One of the things we did in our pre-flight check — every time — was to check the fuel to see if there was water in there. If there was, you had a problem. And aviation fuel is basically kerosene.

So, maybe that prejudiced me against your solution.

• Greg G.

Getting water into your engine’s combustion chamber can lead to throwing a rod through the oil pan, which tends to shorten flight time in planes. I wish to thank all pilots with flight plans over my house for making sure there is no water in the fuel tank.

• Kevin K

I thank them for not flying over my house.

I once had to look for places to ditch when our engine failed (lost about half power) during a training flight. Farmer’s fields do not look friendly; but neither do freeways what with bridges and such. We made it back intact, no ditching required; probably the best lesson I ever got.

• Greg G.

I thank them for not flying over my house.

As long as they fly all the way over my house, I am OK with it. I hate it when they only make it halfway over.

• I think you’re right–my approach had been rather superficial. What if Anne was married … to a woman? That wouldn’t be a real marriage in the eyes of God, of course.

• Lark62

(a – x)(b – x)(c – x) ….. (z – x) = ?

• Greg G.

Zero (0)

• Grimlock

Spoilers for some of the answers, obviously…

1) Yes. There are two possible chains:
(i) Married -> Unmarried -> Unmarried
(ii) Married -> Married -> Unmarried
In either case, the criteria is satisfied.

2) 100 * 110 % = 110, and 110 * 90 % = 99

3) I’m gonna go for option (c). Did some rather sketchy algebra, but I’m not very confident in this answer.

7) Tricky. Need to think about it, and possibly make some assumptions about knowledge. Presumably nobody shall now the true answer afterwards as well?

If so… A says a number that’s less than the genuine wage. Next person adds a part of the wage. Third person adds a part of the wage. Then one more round of adding the remaining part of the wage. Then, assuming they don’t compare notes, they should be in the clear. I think…

8) That’s just mean.

The guests paid \$57. The owner received \$55, and the doorman received \$2. Thus, 57 were given, and 57 were received.

9) Presumably, less than 26 people.

Consider 25 people. Then Joanne won’t get cookies more than once. 24 people? It goes ’round once, and then J takes the last cookie. Awesome. It’s now tempting to think that divisors of 12 will work. If there are eight people around the table, three rounds of eight + 1 for Joanna works. So I feel safe in generalizing to all divisors of 24, i.e. 1, 2, 3, 4, 6, 8, 12.

• Herald Newman

Your answers for number 9 agree with how I understood the problem.

• Grimlock

Oh right. I didn’t wanna spoil myself by reading the comments before answering the puzzles. I can also get behind the reasoning that excludes 1 and 2.

• Kevin K

It took me a lot longer to get there, because I had to break #1 down into smaller steps. My work —

Step 1
Married >> Unknown
Unknown >> unmarried

Step 2 If Unknown = married, then
Married >> married
Married >> unmarried !

Step 3 If Unknown = unmarried, then
Married >> unmarried !
Unmarried >> unmarried.

In both cases, a married person is looking at an unmarried person (assuming for the sake of argument that all of the players here are humans and not dogs or robots or whatever).

• Kev Green

It’s easy to see the answer for 9 if you simplify the problem. Assume instead that there are only 24 cookies and the tray has to end just before it gets to Joanne.

• MR

Yeah, so not 25 because it has to both start and end with Joanne. That was my mistake with that one.

• 3) I found the algebra tricky as well, but I’ve semi-convinced myself that it works out algebraically.

• Grimlock

Yeah, I had another go at it now, this time properly (i.e. with pen and paper and no concrete numbers, just pure algebra), and it worked out rather neatly to option c.

Not going to make any claims to understanding the intuition behind it, though.

• PacMan

The trick is to realise that both cups start with volume x and both end with volume x.
The only way the concentrations could end up different is if the cups ended up with unequal volumes.

• Grimlock

Aah! Yes, that did the trick – thanks!

• I never thought of that. That does seem to be the easy way to see the answer.

• MR

Ok, that got me thinking.

Go grab some checkers or something similar with pieces of two different colors (I’ll use black and white) and separate the colors into two piles (one black, one white) containing equal numbers.

Take any number of white pieces from Pile A and add them to the blacks in Pile B. Now take that same number of pieces from Pile B in any combination of black and white and add them back to Pile A. You will see that no matter how many black pieces you move back to Pile A, the amount of white pieces that you add back will offset the total number in order to balance to the original number you moved in the first place. This “adding back” will automatically balance out the proportions.

For example, if you move 5 white to Pile B, then 3 black to Pile A, you also have to add back 2 whites to Pile A in order to balance to 5. That leaves you with 3 whites in Pile B (5 minus 2) to match your 3 blacks in Pile A. Try any combination, it doesn’t matter. Move a total of four whites and move only one black back, you have to add back 3 of the whites leaving you with 1 white in Pile B (4 minus 3) to match your 1 black in Pile A.

No doubt my explanation will confuse even more, but grab some checkers and you’ll see what I mean. Visually (with discreet units) it makes sense. Visually (with liquids) it’s harder to grasp. Besides, I just kept seeing two cups of coffee with cream in my minds eye when you would really have a cup of coffee with cream, and what would still look like just a cup of milk.

• Interesting approach–move it from the continuous domain to the discrete domain.

Every white piece you move back in step 2 simply reduces the amount of difference, and every black piece you move back equalizes the amount of difference. That sounds convincing to me–thanks.

• MR

Right, then just imagine that as percentages instead of discrete units. Whatever the percentage of one liquid, the other inversely balances.

• Which of course leads us to wonder Who made a reality with these marvelous properties …

• MR

Quit swooning, it’s just simple addition, Bob.

• Grimlock

Whether you give the intuitive answer or analyze further to find the correct answer says something about how you think. This is explored in an article provocatively titled, “3-Question Quiz Predicts Whether You Believe in God.”

Very interesting article!

I wonder how this might impact discussions online about religion. If there are inclinations towards differences in thinking styles, that might make communication a bit more challenging. My general impression is also that counter-apologetics tend to reinforce any analytic inclinations one might have, and could as such lead to increasing the differences.

This sometimes makes me wonder about the morals of trying to make my point by using methods that are not particularly truth-sensitive, such as appeals to emotions. Has anyone else wrestled with this issue?

Grim

• tyler

the solution to #7 is: they discard the anti-worker taboos pounded into them by the bourgeoisie and reveal their salaries to one another, thereby enabling them to support and justify each other in pushing for higher salaries for all of them and potentially uncovering illegal discriminatory disparities, sparking a revolution of the workers against their upper class oppressors

• Kevin K

Adam Ruins Everything did a segment on this.

• Down with the glass ceiling!

• Joe

My answer to all these questions (and more): Jesus.

• Well, yeah. I was going for the other answers.

• wtfwjtd

I think what you meant to say, Bob, was that you were going for the correct answers. But, then again,maybe I think too much 🙂

• Otto

That means that the possible values for the number of people at the table are 25, 13, 9, 7, 5, 4, and 3

This can’t be right because if you had 25 people at the table and 25 cookies, Joanne the cookie hog wouldn’t get 2. If there were 13 people the dish wouldn’t get around twice, if there were 9 it wouldn’t get around 3 times…5 doesn’t work but 4 and 3 do.

(if someone gets on here and proves me wrong my head might just explode)

• JustAnotherAtheist2

FWIW, I was thinking the same thing.

• Just a reminder: the puzzle continues with the Continue Reading button at the bottom. I think some readers missed that.

• RichardSRussell

Problem 7: “Three people with different salaries need to find out their average salary without revealing individual salaries to each other. How?”

They each tell their salary to a trusted 4th person who does the arithmetic and reports the result back to them.

Problem 9: A banker, a tea-partier, and a union member are sitting at a table with a plate of a dozen cookies. The banker takes 11 cookies, then turns to the tea-partier and says “You should watch out for that other guy; I think he wants more than half of your cookie.”

• RichardSRussell

Fun stuff, cleverly presented. Thanks!

• JustAnotherAtheist2

The coffee/milk one didn’t become intuitive for me until after doing the math. I considered each cup to be 100 units and each spoon to be 10 units. The first transfer is easy, it makes it a 90/0, 10/100 split. From there, no matter what ratio the second scoop is it ends up balancing out. If you scoop all coffee, it is 90/10, 10/90. If you could somehow scoop only the original milk, it goes back to 100/0, 0/100. And everything in between works the same.

• Zeropoint

I’ll take a stab at these puzzles. To play fair, I’m going to solve them without looking at any hints or solutions.

1. I expect that I’m missing something, but I’m going with C: Not Enough Information. We know that a married person is looking at Anne, and that Anne is looking at an unmarried person. I don’t see how to resolve this without more information.

2. This is only a puzzle if you don’t walk through it:
100 + 0.1 * 100 = 100 + 10 = 110
110 – 0.1 * 110 = 110 – 11 = 99

3. I’ve seen this one before, so I know what the solution is. They’re equally concentrated; if the volumes start equal and end equal, then the volume of milk missing from the milk cup (i.e. the amount of milk in the coffee) must be exactly equal to the amount of coffee missing from the coffee cup (i.e. the amount of coffee in the milk).

4. The helium balloon will float toward the front of the car, because of buoyancy and the equivalence of gravity and accleration. It floats UP when gravity pulls DOWN, so if the apparent direction of gravity in the car is “down and also back” then the balloon will float “up and also forward”.

5. Mirrors don’t reverse right to left, they reverse front to back. This is easily demonstrated; you can look up proof or work it out for yourself with a model set of coordinate axes or by writing something on a clear piece of plastic or glass.

6. Even crows know how to solve this problem: put something denser than kerosene in the lamp to raise the surface level of the kerosene. Anything denser than kerosene would work, but I imagine that the “right” answer is water from the lake you were just in.

7. Multiple solutions are available. I’d simply write a little bit of Python code to accept three numbers and then compute and display their average, without storing the individual numbers. The three people can look at the code and verify that it meets their secrecy requirements, pass the laptop around, and get the answer. I’m reasonably certain that this isn’t the “correct” answer to the puzzle, though.

8. The \$60 dollar figure is a red herring, only included to confuse you. The guests are out a total of \$57; the hotel has \$55 and the porter has \$2; \$55 + \$2 = \$57.

9. This one is a bit tricky and has lots of opportunities for “off by one” errors to creep in and stack up.

The largest possible number of people at the table is 24. Joanne must have the *first* cookie and the *last* cookie; that’s two cookies, leaving 23 for the other people. 23 plus Joanne is 24.

If the dish goes around twice, then Joanne gets three cookies: the first, the last, and one from the middle. that leaves 22 cookies for the rest of the guests. Since they each get two (one per pass) there must be 11 of them, for a total of 12 at the table.

Three passes: Joanne now gets four cookies. In general, it seems that Joanne gets 2 + (N -1) cookies where N is the number of passes. From this we can conclude that if 25 -(2 + (N -1)) / N is not an integer, then that number of passes is not possible.

Applying 12 lines of Python to automate this process, we have the following list of possible total table members:
1, 2, 3, 4, 6, 8, 12, 24

Edit: Well, now I feel silly. I totally missed the obvious solution to number one, and I made a logic error when I wrote my code. A simpler way to think about it: remove one cookie from the dish, give said cookie to Joanne, and then start it to Joanne’s left with the condition that it must go around ending at Joanne. It should be obvious that the number of people at the table must be a divisor of 24 . . . which is what my list shows? Maybe I made two mistakes that cancelled each other out . . .