Queen's Heads Puzzle

[pdurrant]

You are in a dank, dark, dungeon, awaiting execution for crimes against logic puzzles.

Luckily, you have a chance to redeem yourself. You must play a logic puzzle with the gaoler, and if you win, you go free. The game you must play works like this:

The Gaoler will put a blindfold on you. Then, on a flat table in your cell, the gaoler will place 101 identical coins, which on one side have the Queen's head, and on the other side have the Queen's coat of arms. He will ensure that 21 of the coins are showing the Queen's head, and 80 are showing the Queen's coat of arms.

The challenge you face is to choose 21 coins from the 101 coins on the table by picking them up and placing them flat on the table in front of you, while the other 80 remain in front of the gaoler.

Once you have chosen your 21 coins, and placed them flat on the table in front of you, the gaoler will remove your blindfold. If the Gaoler's 80 coins show more Queen's heads than your 21 coins, your execution will proceed immediately. Otherwise, you go free.

Remember, you are blindfolded, and the pattern on each side of the coins is too similar to tell apart by touch. Oh — and you only have five minutes to make your choices.


1. What strategy maximises your chance of gaining your freedom?
2. Using this strategy, how likely are you to gain your freedom?

As usual, answers in

Spoiler:

spoiler tags

please, so everyone can have a go.

(I've seen this puzzle in a couple of places, but most recently here: http://www.nickyee.com/index-puzzles.html )

[happy_terd]

Spoiler:

I would pick 21 coins and flip the coins over when I placed them on the table in front of me. If I pick a coin that is a coat of arms then it turns into a head it adds towards reaching a tie with the Gaoler's 21 which I need to at least tie. If I pick up a coin that is a head then flip it I take 1 of the heads I need to match away from Gaoler and still end up in at least a tie.

Just reread my logic. I believe I am correct. I may need to explain it better. Let me know if what I said makes sense.

[glowhappy]

Spoiler:

1. just pick up coins at random and turn it over so that the opposite side is faced up.
Since you have 21 chances its more likely that of the remaining 80 coins, you will have more of the Queen's Heads side up because if you pick up a Coat side, you'd have flipped it and ensured you have a Head side. If you pick up a Head coin, then you've lessen the Goaler's Head coin count.

2. using the method above, the ratio of Head to Coat coins will stay relatively the same b/c flipping the coin will result in it being either one or the other. Since there were more Coat coins to begin with, roughly 80 percent, then your chances of choosing a Coat coin and flipping it into a head coin 8 of 10. Ensuring that you will indeed have more Head coins.

*Each time I pick up a Coat coin and flip it, I will have a Head coin and have more or be equal to count with Goaler. Each time I pick up a Head coin and flip it I would have lessen the the Goaler's Head count, therein still tie with him in Head count.

I think so do I die?

[happy_terd]

Originally Posted by happy_terd

Spoiler:

I would pick 21 coins and flip the coins over when I placed them on the table in front of me. If I pick a coin that is a coat of arms then it turns into a head it adds towards reaching a tie with the Gaoler's 21 which I need to at least tie. If I pick up a coin that is a head then flip it I take 1 of the heads I need to match away from Gaoler and still end up in at least a tie.

Just reread my logic. I believe I am correct. I may need to explain it better. Let me know if what I said makes sense.

Just tested this out with real coins. I am correct.

You are also correct glowhappy.

[jbjb]

Spoiler:

You can be certain of going free.

Pick 21 coins at random (you have no choice, assuming they are randomly positioned by the gaoler), and turn each one over before positioning it on your table. You will end up with the same number of Queen's heads as your gaoler.

Reasoning:

At the start, the gaoler has 21 more heads than you (he has 21 and you have none, as you have no coins).

Each time you pick a coin either it was a head on his table or not.

If it was a head on his table, you've turned it over to a coat of arms on your table - you've reduced his number of heads by one and made no difference to your number of heads.

If it was a coat of arms on his table, you've turned it over to a head on your table - increasing your count of heads by one and making no difference to his count.

I.e. each coin you move in this way will reduce the excess number of heads he has over you by one, irrespective of what the coin initially showed.

So, after you've moved 'n' coins this way, he will have '21-n' heads more than you. Once you've moved your chosen 21 coins, you and he will have the same number of heads. Hence, he won't have more heads than you and you therefore you go free

/JB

[pdurrant]

Originally Posted by happy_terd

Spoiler:

I would pick 21 coins and flip the coins over when I placed them on the table in front of me. If I pick a coin that is a coat of arms then it turns into a head it adds towards reaching a tie with the Gaoler's 21 which I need to at least tie. If I pick up a coin that is a head then flip it I take 1 of the heads I need to match away from Gaoler and still end up in at least a tie.

Just reread my logic. I believe I am correct. I may need to explain it better. Let me know if what I said makes sense.

Yes, you have it right. Congratulations.

Spoiler:

In fact, using the method you propose (which is the right one) you always tie with the Gaoler, and so go free. It's not possible with this method to end up with more heads than the gaoler.

[pdurrant]

Originally Posted by glowhappy

Spoiler:

1. just pick up coins at random and turn it over so that the opposite side is faced up.
Since you have 21 chances its more likely that of the remaining 80 coins, you will have more of the Queen's Heads side up because if you pick up a Coat side, you'd have flipped it and ensured you have a Head side. If you pick up a Head coin, then you've lessen the Goaler's Head coin count.

2. using the method above, the ratio of Head to Coat coins will stay relatively the same b/c flipping the coin will result in it being either one or the other. Since there were more Coat coins to begin with, roughly 80 percent, then your chances of choosing a Coat coin and flipping it into a head coin 8 of 10. Ensuring that you will indeed have more Head coins.

I think so do I die?

You certainly have the right strategy. You haven't analysed the outcome sufficiently yet though. You've answered Q1 but not Q2.

[pdurrant]

Originally Posted by jbjb

Spoiler:

You can be certain of going free.

Pick 21 coins at random (you have no choice, assuming they are randomly positioned by the gaoler), and turn each one over before positioning it on your table. You will end up with the same number of Queen's heads as your gaoler.

Reasoning:

At the start, the gaoler has 21 more heads than you (he has 21 and you have none, as you have no coins).

Each time you pick a coin either it was a head on his table or not.

If it was a head on his table, you've turned it over to a coat of arms on your table - you've reduced his number of heads by one and made no difference to your number of heads.

If it was a coat of arms on his table, you've turned it over to a head on your table - increasing your count of heads by one and making no difference to his count.

I.e. each coin you move in this way will reduce the excess number of heads he has over you by one, irrespective of what the coin initially showed.

So, after you've moved 'n' coins this way, he will have '21-n' heads more than you. Once you've moved your chosen 21 coins, you and he will have the same number of heads. Hence, he won't have more heads than you and you therefore you go free

/JB

Exactly right. Well done.

[obs20]

Spoiler:

It seems that I have to place at least 11 heads in front of me to survive. Since on the table there are 80 tales and 21 heads which is a ratio of about 4 to 1. What makes sense to me is to take each of the 21 coins I pick and turn them over before placing them back down on the table in front of me.

[omk3]

Spoiler:

I pick up my 21 coins at random. I have x heads, and 21-x tails.
The gaoler has 21-x heads.
If I flip all my coins over, I now have 21-x heads, and x tails.
So I will always have the same number of heads as the gaoler, and always go free. Yay!

[pdurrant]

Originally Posted by obs20

Spoiler:

It seems that I have to place at least 11 heads in front of me to survive. Since on the table there are 80 tales and 21 heads which is a ratio of about 4 to 1. What makes sense to me is to take each of the 21 coins I pick and turn them over before placing them back down on the table in front of me.

That sounds like the right strategy, to answer Q1. How about Q2?

[pdurrant]

Originally Posted by omk3

Spoiler:

I pick up my 21 coins at random. I have x heads, and 21-x tails.
The gaoler has 21-x heads.
If I flip all my coins over, I now have 21-x heads, and x tails.
So I will always have the same number of heads as the gaoler, and always go free. Yay!

Exactly right, and nicely explained.

[obs20]

Originally Posted by pdurrant

That sounds like the right strategy, to answer Q1. How about Q2?

I agree with jbjb his explanation is much better than anything I could come up with.

[Mike L]

Originally Posted by happy_terd

Just tested this out with real coins.

That's a bit dangerous, isn't it?

[happy_terd]

Originally Posted by Mike L

That's a bit dangerous, isn't it?

I live on the edge.