Riddle Answers

Thanks to all who commented on the riddles yesterday. The first riddle had the highest number of responses to any post I’ve made to date. Here’s my proposed answer to the riddles (paradoxes?) posted yesterday, here and here.

Starting with Riddle 2-A. In brief:

Suppose we are presented with the opportunity to open our wallets. Whoever has more money has to give it to the other guy.

A simple analysis suggests that you have a 50/50 chance of winning, and if you do, you’ll gain more money than when you lose.

I think this is the easiest to address:

Clearly, no new money is being generated in the interaction, so there can be no overall positive expectation for playing the game (assuming equal information between the players).

If in fact, this game DID generate positive total expecations, then I would encourage all married individuals to go right now to their spouses and play this game – the two of you will come out ahead, in the aggregate.

Or not. Anyways, I could run a simulation to address this problem, but I think it’s simple enough that it’s unnecessary. I won’t present math to try to analyze WHY this apparent paradox exists, but follow the Wikipedia link at the end to those who do.

2-B is basically the same as 2-A, except that rather than their being an unknown difference between the two player’s wallets (envelopes), it’s known that one player has twice the other player’s money. I’ll skip it, because I think I address the case better in analyzing Riddle 1, below.

Riddle 1. In brief:

An eccentric billionaire invites to participate in a challenge.

He has taken a number of cards, and had a number printed on each of them. On the reverse side of the card, there is another number – twice the value of whatever is on the front.

The cards are put in a bag. You draw one, and see that it shows 53. By prior agreement, he will pay you that sum (in dollars), or allow you to pay a 10% premium (i.e. $5.30) to flip the card over and accept whatever sum is on the back, instead.

Do you pay to flip the card?

The key here is that, although it APPEARS that the amount of money on the other side of the card has a 50/50 chance of being twice or half what’s on the side you see, it’s not quite that simple. In fact, the cases where the amount on the other side is higher are disproportionately drawn from the set of card face values with low values on them. Confusing? Sure, but it’s easy to simulate:

Here’s the code to test this. It sets up a simulation where 10 cards receive a random value from 1 to 100, and the back sides receive double that amount. Then I draw a card and record the results. I repeat the experiment 1,000,000 times (with a new, random set of cards each time), and here are the results:

Flip mean: 68.35
No-Flip mean: 75.53
Flip advantage (penalty) -9.51%
Flip winners percent: 50.34%
Flip winners mean (starting): 50.51
No-Flip winners mean (starting): 100.89

By flipping, you end up with a 9.51% overall penalty – very close to the 10% penalty you’d expect (i.e. the cost of the flip.)

Note that the mean result from not flipping is a bit over 75. That’s the average of the front side (1..100, average = 50.5), and the back (2..200, average = 101), total average = (50.5+100)/2 = 75.25.

Note that on the 4th line, we see that in this simulation, the odds of ‘flip’ improving your position on any ONE bet were 50.34% – close to the expected value of 50%.

However, the next two lines show why the whole paradox exists: The occasions when ‘flip’ works are those where the starting value of the card (i.e. the number you see) are relatively low (mean 50.51), and the no-flip works on higher relative values (mean 100.89). So even though, on any SINGLE observation, flipping should work half the time, the problem is that those observations cluster into the group where the initial observed value is low.

I also did the same test altering one aspect of the simulation. What if, rather than the ‘flip’ option giving you whatever is ALREADY on the other side of the card, it gives you a random (50/50) chance at double or half your money. Here’s the results:

Flip mean: 87.41
No-Flip mean: 75.53
Flip advantage (penalty) 15.72%
Flip winners percent: 50.47%
Flip winners mean (starting): 75.54
No-Flip winners mean (starting): 75.52

Flipping here improves your outcome by about 15%, as expected. The key difference? The starting values of the flip winners and the no-flip winners are effectively the same (~75). So flipping is not paying off only in the low value cases, but rather, in an even spectrum across all cases.

Some might argue that in the original problem, the eccentric billionaire’s range of possible numbers for the cards was unstated, and in my simulation, I’ve arbitrarily chosen 1-100. Fair enough. Here’s the results on two other ranges.
(All using the ‘preset deck’ scenario)

Card Range 1-20,000:
Flip mean: 11636.22
No-Flip mean: 12858.08
Flip advantage (penalty) -9.50%
Flip winners percent: 50.36%
Flip winners mean (starting): 8596.28
No-Flip winners mean (starting): 17180.83

Card Range 1524-7621 (i.e. a fairly arbitrary low and high end)
Flip mean: 6006.02
No-Flip mean: 6637.75
Flip advantage (penalty) -9.52%
Flip winners percent: 50.35%
Flip winners mean (starting): 4437.15
No-Flip winners mean (starting): 8869.06

Note that I couldn’t easily test with max values above 32,768, because of limitations of my random number generator. But if you still have doubts, download the code, put in a better random number generator, and test from 1 – 1 billion. The results will be effectively the same.

Finally, I know this is only a simulation, and doesn’t include a good mathematical explanation of WHY this works in such a counter-intuitive manner. The math of this problem is beyond me, but I’d refer you to the Wikipedia article on this subject, with many links to mathematical analyses at the bottom.

38 Responses to “Riddle Answers”

  1. Esteban Says:

    I assume that the player in your game don’t know the range of values. Otherwise, your simulation has the flaw that it doesn’t take trivial decisions into account (AFAICT). The number 1 is a trivial flip decision, and all numbers above 100 are trivial no-flips. The simulation shouldn’t decide at random in these cases.

    What if the range of values was known, and the additional rule was added that whenever you pick a card by a side with a trivial value, you must toss the card back into the bag and pick again? Would this make all nontrivial values a real 50/50 expected chance to win or lose? Would this change the statistical results of the simulation? At least it would eliminate the complication that the player may speculate with the range of possible values.

  2. Ron Gilbert Says:

    I agree with the results from a math point of view, but, we don’t get to play this game over a long period of time, we just get to play it once. So, my decision to flip or not flip is going to be based on my being happy with the amount shown on the card. If it says $1,000,000, I’m going to take it. I’m happy with that amount of money. If it says $100, what the hell, I’ll risk it. It’s kind of like when you play the lottery. You’re chances of winning are close to zero, but it’s only a buck and it’s kind of fun to check your numbers and…someone does win.

    I know this wasn’t what you were asking, but it does figure into someone decision.

  3. Phil Steinmeyer Says:

    Hmm, WordPress is not very good at formatting code. They have a CODE tag, but it doesn’t really work. Here’s a link to the code as ASCII text.

  4. Phil Steinmeyer Says:

    Esteban – as I note at the end of my post, it doesn’t matter what the (unstated) range is – the results are the same.

    Obviously, if the player KNOWS the range in advance, then he can change his play strategy to profit quite nicely from flipping. But assuming the player does not know the range (as stated in the original riddle), then flipping is a net negative (I guess an exception can be carved out if the value is ’1′, but that doesn’t substantially change the overall conclusion – it’s just an artifact of the rounding process…)

  5. MrPhil Says:

    Isn’t this like the old three door arguement from Parade’s ask Marilyn? That you should take all the chances you can, because you started with nothing and you have nothing to lose?

  6. Patricia Says:

    A man rides into town on Friday. He stays three nights and leaves on Friday.how did he do it

  7. Phil Steinmeyer Says:

    Friday is the name of his horse?

  8. Pat Says:

    there are 3 men in the ocean, and they all need to get back to shore. If the boat can only hold 300lbs, how do all the men get back to shore, using the boat, if on man weighs 300 lbs and the other 2 men weigh 150 lbs each?

  9. Phil Steinmeyer Says:

    Well – two of the men could float alongside the boat, holding onto the edge while the 3rd man paddles the boat…

  10. Eli Says:

    I would like to know these riddles

    What gets wetter and wetter, the more it dries?

    Give me food, and I will live; give me water, and I will die. What am I?

    The beginning of eternity, The end of time and space, The beginning of every end, And the end of every place.

  11. Phil Steinmeyer Says:

    Wetter as it dries – towel

    Food – not sure

    beginning, end, etc. The letter ‘e’

  12. Devin Says:

    i need a answer to a riddle…

    Dont let the name fool you, the rockies aint the stones, you’d have to get over at this cow country clash. What is it?

  13. jennifer green Says:

    In the jones family each daughter has the same number of brothers as she has sisters. Each son has twice as many sisters as he has brothers. how many sons and daughters are in the family?

  14. Phil Steinmeyer Says:

    3 sons, 4 daughters

  15. kristeen Says:

    I am round like a ball but won’t bouce or roll. I have no wings but I know how to fly. I am not made of glass but I break easily and you can see through me.

    What am I?!

  16. Phil Steinmeyer Says:

    A soap bubble?

  17. Alex Major Says:

    the beginning of eternity, the end of time and space, the beginning of every end, and end of every place! answer this asap please!!!

  18. Phil Steinmeyer Says:

    the letter ‘e’

  19. Mom Brady Says:

    If you count 20 houses on your right going to school, and 20 houses on your left coming home, how many houses in all have you counted?

  20. Phil Steinmeyer Says:

    Well, it would be the same 20 houses (when you return home, your left side is the side that would have been your right on the way there).

  21. John Barrett Says:

    there are 3 men in the ocean, and they all need to get back to shore. If the boat can only hold 300lbs, how do all the men get back to shore, using the boat, if on man weighs 300 lbs and the other 2 men weigh 150 lbs each?

    The 2 men that weigh 150lbs each go to shore. One gets out. The other goes back and gets out. The 300 lbs guy goes to shore and gets out. The first 150 lb guy gets in and gets the other 150lb guy.

  22. Pamela Says:

    There are three words in the English language that end in “gry”. ONE is angry and the other is hungry. EveryONE knows what the third ONE means and what it stands for. EveryONE uses them everyday, and if you listened very carefully, I’ve given you the third word. What is it?

  23. Phil Steinmeyer Says:

    There are no other COMMON English words ending in gry – only some obscure stuff like aggry or meagry, and other very obscure/archaic words…

  24. Pamela Says:

    “gry” words
    Thank you for the reply. Of course, I could not accept that and so I wasted hours and hours researching – only to discover that the first person I asked (you) was correct. If anyone cares to delve further, below is the info I found on the Tempe Public Library website. Thank you again.

    Those Dreaded “-gry” Words
    For years and years and years, librarians have been fielding questions such as, “I heard there are three commonly used English words ending in ‘-gry’. I know about ‘hungry’ and ‘angry’, but what is the other word?”

    This question is asked many, many times, every month, every year, and has been asked for as many years as anyone who works at this particular library can remember. If we seem to you be somewhat less than thrilled to hear the “-gry” riddle, it is because, to use a massive understatement, the novelty has worn off. So far, we have found many answers, the two most common of which we’ve posted below.

    That being said, please understand that “words ending in gry” is the one topic on which Tempe Public Library does not respond to email. Here is one reason, quoted from the Frequently Asked Questions file of the rec.puzzles newsgroup: “There are many generally unsatisfying “trick” answers to the problem, which depend on a specific wording of the question or that the question be spoken instead of written. There seems to be no agreement among puzzle historians about which form is the original, or even the age of the problem. In any event, it is apparent that the frequent mutations of the puzzle statement over the years have erased whatever answer was intended by the original author.” The other reason we don’t answer email about -gry words is that we are already busy providing library service for the people who live in Tempe, Arizona. For two answers to the riddle, just read on:

    Answer Number One:

    Here one answer, in a nutshell, quoted from “Gry, Gry, Everywhere, and Not a Clue In Sight”, from The Word Detective, the online version of Words, Wit and Wisdom, a newspaper column that answers readers’ questions about words and language, and is currently syndicated in newspapers in the U.S., Mexico and Japan. Here is the gist of what its author has to say:

    Perhaps the whole puzzler is more a grade school antic than anything else. The way I heard the setup for the question was this:

    There are three words in the English language that end with “gry.” One is hungry and the other is angry. What is the third word? Everyone uses this word every day, everyone knows what it means, and knows what it stands for. If you have listened very closely I have already told you the third word.

    If you read the second sentence you see that the “third” word is “hungry”.

    The author is writing here about the third word in the second sentence of the riddle, exactly as quoted, NOT some mythical third commonly used English word ending in “-gry”. We admit this is a rather stupid riddle, but then we we didn’t make it up; we just answer it, over, and over, and over.

    The Word Detective may be found on the web at http://www.word-detective.com.

    Answer Number Two:

    The other answer also involves the way in which the riddle is posed, and will work only when spoken. It goes like this: “There are at least three commonly-used English words ending in ‘g’ or ‘y’. One is ‘hungry’, the other is “angry’…” and so on. After the victim has been given time to be driven sufficiently nuts, the riddler is supposed to reveal the magic word, ‘or’, which, of course, sounded like ‘r’ in the spoken riddle. This one was reported in Marilyn Vos Savant’s column, “Ask Marilyn”, in the Parade supplement to the March 9, 1997 issue of the Arizona Republic.

    …And For the Terminally Curious:

    Click here to see an exhaustive list of un-common English words ending in “-gry”, posted in the rec.puzzles archive.

    Last updated 16 September 2002; joke still old in August 2004.

  25. John H Says:

    I turn polar bears white
    and I will make you cry.
    I make guys have to pee
    and girls comb their hair.
    I make celebrities look stupid
    and normal people look like celebrities.
    I turn pancakes brown
    and make your champagne bubble.
    If you squeeze me, I’ll pop.
    If you look at me, you’ll pop.
    Can you guess the riddle?

  26. Phil Steinmeyer Says:

    Hmm – no idea…

  27. CensusTaker Says:

    The first one isn’t really a riddle, its a maths/logic puzzle- and it just depends on how much money you’ve got on you compared to whatever the average is (which would be pretty impossible to calculate) to determine the real odds. From a gambling perspective it would indeed just come down to how much you were willing to lose.

    The 2nd one is a no brainer… if the numbers are whole numbers- intergers- that is, which I assume they would be in the original wording of the Riddle rather than this summary (Riddlers, from the Sphinx to Edward Nigma are nearly always sticklers for semantics) if the number you draw is 53, an odd number, then the number on the other side would have to be double not half of 53. If they’re not whole numbers, well, thats when the math plays a part.

    As for the other unanswered riddles in this page:

    Fire lives when you give it food and dies if you give it water.

    And the final one is `Yes’. Because Im that good.


  28. CensusTaker Says:

    Hey the first time I hit the link to find what the original question was, it didn’t work, but I just double checked after my foolish post and it worked that time, so now i see that im wrong with the second one. Should have know a logic puzzle worth posting on the net wouldn’t be that simple. Still, Id flip the card, you’ve got more to gain than to lose.

  29. Candy Says:

    here is the question that i have i need an answer…..well actually a riddle

    I turn polar bears white
    and I will make you cry.
    I make guys have to pee and girls comb their hair
    I make celebrites look stupid and normal people look like celebrities
    I turn pancakes brown
    and make your champane bubble
    If you squeeze me i’ll pop
    If you look at me you’ll pop

    What am I??

    I really need help figureing out this riddle

  30. Phil Steinmeyer Says:

    Candy – I couldn’t figure out the one you posted, and did an internet search for it. Here’s a discussion I found:


    (Hopefully the link works).

    Short summary – there really isn’t a satisfying answer to the riddle. “Pressure”, “Time”, and “Salt” were all proposed, but none are perfect, IMO. Also, in some versions, the riddle ends something like “Can you guess what I am?”, and in that case, the answer is supposedly “No”. I’m not thrilled with any of those answers though.

  31. john Says:

    a certain lock has no combination or key and does not stop people from getting the object inside it. Its owners are pleased with its performance and make no effort to repair or replace it.

  32. miataman Says:

    What thing exists in life but cannot been seen?

    need help on this one…

  33. I Derno Says:

    what thing exists in life but cannot be seen?

    a lie??

    i need help with johns

  34. curious georgina Says:

    is it a locket?

  35. qaidi Says:

    There are 4 men who want to cross a bridge. They all begin on the
    same side. You have 17 minutes to get all of them across to the other
    side. It is night. There is one flashlight. A maximum of two people
    can cross at one time. Any party who crosses, either 1 or 2 people,
    must have the flashlight with them. The flashlight must be walked
    back and forth, it cannot be thrown, etc. Each man walks at a different
    speed. A pair must walk together at the rate of the slower man

    Man 1: 1 minute to cross
    Man 2: 2 minutes to cross
    Man 3: 5 minutes to cross
    Man 4: 10 minutes to cross

    For example if Man 1 and Man 4 walk across first, 10 Minutes have
    elapsed when they get to the other side of the bridge. If Man 4
    returns with the flashlight, a total of 20 minutes have passed, and
    you have failed the mission

  36. Phil Steinmeyer Says:

    1 and 2 cross (2 minutes elapsed)
    1 returns (2+1=3 minutes elapsed)
    3 and 4 cross (3+10 = 13 minutes elapsed)
    2 returns (13+2 = 15 minutes elapsed)
    1 and 2 cross (15+2 = 17 minutes elapsed)

  37. Jack Says:

    “A blind man sees a rabbit, a one legged man runs him down and a naked man picks him up and puts it in his pocket.” I’ve been trying to solve this one for some time. The riddle was told to me so the word ‘sees’ might also be ‘sieze’. The same answer explains all three and its something that you hear every day.

    Any ideas?

  38. Phil Steinmeyer Says:

    Hmm, I can’t figure that one out Jack. A quick internet search did not find me an answer either.

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