Brain Teasers

[loseyourname]

Is the next number 54?

Although Seapahn seems to think that's a valid answer, I don't agree. You'll find that 1 and 10 add up to 1, whereas 28 adds up to 10. If you just add up the digits, you get this:

1, 3, 4, 9, 1, 3, 4, 9, 10, 3, 4, 9, 10, 12, 4

Of course if you further add up the multidigit numbers in the second sequence, you will then get:

1, 3, 4, 9, 1, 3, 4, 9, 1, 3, 4, 9, 1, 3, 4

In which case a pattern becomes clear. There is a simpler pattern, though, and I think this is what Seapahn was expecting.

Start with the number 1. Multiply it by 3. Then add 1. You have the first three digits: 1, 3, 4. Multiply 3 by 3, then add 1. You have the next two: 9, 10. Then multiply 4 by 3, then add 1. You now have the numbers 1, 3, 4, 9, 10, 12, 13. Now multiply 9 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28. Now multiply 10 by 3, then add 1. You have 1, 3 ,4 , 9, 10, 12, 13, 27, 28, 30, 31. Now multiply 12 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37. Finally, you do this with 13 (multiply by 3, then add 1) to get the full sequence originally given: 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40. The next number to be multiplied in this sequence would be 27, so the answer is

81.

[loseyourname]

How will that one person immediately know that it's him if he has never seen his own eye color nor was he told by others?

If he's the only one, he will know because he knows that everyone else has brown eyes. Prior to the goddess's announcement, he might have had brown eyes as well. Now he knows that he does not.

[loseyourname]

I don't believe in slapping others, just my forehead. I do believe in other sorts of punishments. One of them involving a brush and wet tiles. Can you guess what it is?

Making someone scrub your bathroom floor?

[anileve]

If he's the only one, he will know because he knows that everyone else has brown eyes. Prior to the goddess's announcement, he might have had brown eyes as well. Now he knows that he does not.

There is a certain island where all of the inhabitants have either brown or blue eyes. It's a small island, and everybody knows everyone else, so everybody knows everyone else's eye color.

The wording indicates that there is more than one person on the island. Therefore, if more than one person resides on the island it's impossible for him to know that he has blue eyes unless someone tells him. The problem doesn't say that the goddess points out the blue eyed person, but if it did, then the answer would be obvious, because once the person learns of his eye color he must commit suicide. Your explanation is not very clear.

[anileve]

Making someone scrub your bathroom floor?

Correct!!! That was the toughest one out of all; you are special.

[anileve]

Although Seapahn seems to think that's a valid answer, I don't agree. You'll find that 1 and 10 add up to 1, whereas 28 adds up to 10. If you just add up the digits, you get this:

1, 3, 4, 9, 1, 3, 4, 9, 10, 3, 4, 9, 10, 12, 4

Of course if you further add up the multidigit numbers in the second sequence, you will then get:

1, 3, 4, 9, 1, 3, 4, 9, 1, 3, 4, 9, 1, 3, 4

In which case a pattern becomes clear. There is a simpler pattern, though, and I think this is what Seapahn was expecting.

Start with the number 1. Multiply it by 3. Then add 1. You have the first three digits: 1, 3, 4. Multiply 3 by 3, then add 1. You have the next two: 9, 10. Then multiply 4 by 3, then add 1. You now have the numbers 1, 3, 4, 9, 10, 12, 13. Now multiply 9 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28. Now multiply 10 by 3, then add 1. You have 1, 3 ,4 , 9, 10, 12, 13, 27, 28, 30, 31. Now multiply 12 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37. Finally, you do this with 13 (multiply by 3, then add 1) to get the full sequence originally given: 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40. The next number to be multiplied in this sequence would be 27, so the answer is

81.

Brilliant! I would have gotten lost in the addition.

[loseyourname]

The wording indicates that there is more than one person on the island. Therefore, if more than one person resides on the island it's impossible for him to know that he has blue eyes unless someone tells him. The problem doesn't say that the goddess points out the blue eyed person, but if it did, then the answer would be obvious, because once the person learns of his eye color he must commit suicide. Your explanation is not very clear.

Think about it this way. Put yourself in the shoes of the islander. You know that everyone on the island, other than you, has brown eyes. You don't know your own eye color, though, so it may be brown or blue. When the goddess says that someone in the crowd has blue eyes, you immediately know it can't be any of the other islanders, because you know that all of them have brown eyes. You then know that you must be the person with blue eyes.

[anileve]

Process of elimination I guess. Thanks, I got it! Took me long enough.

[Anonymouse]

I must admit, even I had trouble with that riddle and it took me way longer to get, with some tips, before finally looking for the answer. I spent a month thinkin about this, since I am slow when it comes to analytical shyt. I couldn't hold out anymore. Thus here is the way it goes. I had trouble understanding why everyone dies. Here's the best way I've found to explain it. People seem to get right away why everyone dies if there is only one person with blue eyes. So we have rule 1.

Rule 1: If only one person on the island has blue eyes, he will die within one day. Now, suppose there are two people with blue eyes. Each of them knows that one of the following is true:
a) I have blue eyes, and the total number of people with blue eyes is 2.
b) I have brown eyes, and the total number of people with blue eyes is 1.

By Rule 1, they know that if there is only one person with blue eyes, he will die within one day. After one day elapses, they know that option b) above is false, since the other guy didn't die. Therefore, they can deduce that they have blue eyes, and kill themselves. This gives us rule 2.

Rule 2: If two people have blue eyes, they will die on the second day. Now, suppose there are three people on the island with blue eyes. Each of them knows that one of the following is true:
a) I have blue eyes, and the total number of people with blue eyes is 3.
b) I have brown eyes, and the total number of people with blue eyes is 2.

By Rule 2, they know that if option b) is true, the blue-eyed folks will die on the second day. When the second day passes, and nobody dies, they realize option b) is false, option a) is true, they have blue eyes, and they all die on the third day, giving us rule 3.

Rule 3: If three people have blue eyes, they will die on the third day. And on. If there are N people with blue eyes, they will all die on the Nth day. After they do, all the remaining people can deduce that they have brown eyes and die the day after.

[Sip]

Start with the number 1. Multiply it by 3. Then add 1. You have the first three digits: 1, 3, 4. Multiply 3 by 3, then add 1. You have the next two: 9, 10. Then multiply 4 by 3, then add 1. You now have the numbers 1, 3, 4, 9, 10, 12, 13. Now multiply 9 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28. Now multiply 10 by 3, then add 1. You have 1, 3 ,4 , 9, 10, 12, 13, 27, 28, 30, 31. Now multiply 12 by 3, then add 1. You have 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37. Finally, you do this with 13 (multiply by 3, then add 1) to get the full sequence originally given: 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40. The next number to be multiplied in this sequence would be 27, so the answer is

81.

Very good!!!! The technical explanation for the pattern is "All positive binary integers assumed to be in base 3, converted to decimal." But the effect is basically what you found so you don't need to know about different bases to solve this although it certainly helps.