[Guest]

You and your two best friends have numbers written on your foreheads. You are told that the three numbers are primes and that they form the sides of a triangle with prime perimeter. You see 5 and 7 on your friends' forehead, both of whom state that they cannot deduce the number on their own foreheads. What number is written on your forehead?

[Guest]

11?

i guess 5 and 7 will be correct too

[grey cells]

By Heron's formula ,

Perimeter , s=a+b+c.

a,b,c=sides of triangle=numbers written on the foreheads.

s=a+5+7.

s=a+12.

a=5(or)7.I think numbers with more than one digit are not allowed.

I am not sure about the answer though.

[roolstar]

But it seems to me that neiher can you!!

You can either have 1, 5, 7 or 11 written on your forehead. Any of these numbers would leave your friends uncertain about thier numbers!

Am I off here?

Edited April 7, 2008 by roolstar

[Guest]

But it seems to me that neiher can you!!

You can either have 1, 5, 7 or 11 written on your foreheads. All these would leave your friends uncertain about thier numbers!

Am I off here?

1 is neither prime nor composite

[roolstar]

I think numbers with more than one digit are not allowed.

You cannot go above 12 for obvious reasons... but you can be 11!

[grey cells]

You cannot go above 12 for obvious reasons... but you can be 11!

Sorry , but I cannot get your obvious reasons .

[Guest]

You cannot go above 12 for obvious reasons... but you can be 11!

Ya that is true ... the possible numbers will be 3,5,7 and 11. But only one of them gives a unique solution ...

[Guest]

how can 3 be a possible no.? if added to 5 and 7, it gives 15 which isn't prime

[grey cells]

Ya that is true ... the possible numbers will be 3,5,7 and 11. But only one of them gives a unique solution ...

Brhan , I cannot see how 3 works and can you please explain why numbers above 12 are not possible.

I am stumped here.

[Guest]

how can 3 be a possible no.? if added to 5 and 7, it gives 15 which isn't prime

That is true ... what I mean is it is a prime number.

[Guest]

Brhan , I cannot see how 3 works and can you please explain why numbers above 12 are not possible.

I am stumped here.

Alright GC,

Of course 3 is not a solution. Any number above 12 is not possible because there is a property of triangles in which the sum of the two sides is greater than the third side. So, if a, b and c are the sides of a triangle then a+b > c; a+c > b and b+c > a. Since 5 and 7 are already given and 5+7=12 the third side must be less than 12.

So, the possible solutions are 5, 7 and 11. But only one of them can give a unique solution ... and that is what I am looking for.

[grey cells]

Alright GC,

Of course 3 is not a solution. Any number above 12 is not possible because there is a property of triangles in which the sum of the two sides is greater than the third side. So, if a, b and c are the sides of a triangle then a+b > c; a+c > b and b+c > a. Since 5 and 7 are already given and 5+7=12 the third side must be less than 12.

So, the possible solutions are 5, 7 and 11. But only one of them can give a unique solution ... and that is what I am looking for.

Thanks , Brhan . I should definitely refresh my geometry .

[grey cells]

1st friend(5) : Sees 7 and my number(5,7 or 11).

If he sees 5 , his could be 5,7 or 11.

If he sees 7 , his number is 5

If he sees 11 , his number could be 5,11 or 13.

2nd friend(7) : Sees 5 and my numbers(5,7 or 11).

If he sees 5, his number is 7.

So 5 and 7 are eliminated.

So my number is 11.

Please verify it Brhan.

Edited April 7, 2008 by grey cells

[Guest]

So, the possible answers are (initially) 3, 5, 7, 11 since those are the only prime numbers that would satisfy the triangle inequality theorem. 3 is out, since 3+5+7 = 15 (not prime), but the other 3 are possible answers.

I'm stumped after that. Usually, its the clue that since the other people can't figure it out but you can lets you know what to do next. But with the three scenarios, here's what I see and why I'm stumped.

If you have a 5...

...then the other guy who has a 5 is seeing a 5 and a 7, and sees 3 possible prime totals (17, 19, 23) and can't answer.

...and the other guy who has a 7 is seeing a 5 and a 5, and sees 2 possible prime totals (13 and 17) and can't answer.

If you have a 7...

...then the other guy who has a 5 is seeing a 7 and a 7, and sees 2 possible prime totals (17 and 19) and can't answer.

...then the other guy who has a 7 is seeing a 5 and 7, and sees 3 possible prime totals (17, 19, 23) and can't answer.

If you have an 11...

...then the other guy who has a 5 is seeing a 7 and 11, and sees 3 possible prime totals (23, 29, 31) and can't answer.

...then the other guy who has a 7 is seeing a 5 and 11, and sees 2 possible prime totals (23 and 29) and can't answer.

If 2 of these 3 scenarios had someone who could figure it out, then you would choose the remaining one. If even one of them had someone, I might have an idea. But I'm stumped. No matter what number you have, none of the others should be able to figure it out, so that clue doesn't seem to help (initially, at least).

[Guest]

GREYCELLS SAYS:

1st friend(5) : Sees 7 and my number(5,7 or 11).

The first friend has no idea of his number, so has no idea of your thought process, so has no idea that you are limited to 5, 7 or 11. Plus, he's not limited to those either. It's quite possible that 3 or 13 are feasible choices for some scenarios.

[Guest]

1st friend(5) : Sees 7 and my number(5,7 or 11).

If he sees 5 , his could be 5,7 or 11.

If he sees 7 , his number is 5

If he sees 11 , his number could be 5,11 or 13.

2nd friend(7) : Sees 5 and my numbers(5,7 or 11).

If he sees 5, his number is 7.

So 5 and 7 are eliminated.

So my number is 11.

That is perfect. Well done Grey Cells. Nayama has got it as well, though without reasoning. Nice one Nayama ...

Edited April 7, 2008 by brhan

[Guest]

That is perfect. Well done Grey Cells. Nayama has got it as well, though without reasoning. Nice one Nayama ...

I'm either missing something, or there's something missing from the question.

What this answer is saying is that if the person with the 5 sees two 7's, then he knows he has a 5, because 5+7+7 = 19, and 7+7+7=21 (composite) and 11+7+7 = 25 (composite).

But why can't this person think he has a 3? 3+7+7 = 17 (also prime).

While YOU are limited into thinking you have only a 5, 7, or 11 based on the two numbers YOU see, there's no reason that I can see that the other two are limited. There's no reason for THEM to eliminate 3 or 13 as possibilities if they see two numbers on your head and on the remaining person's head that would satisfy the triangle inequality.

[Guest]

That is perfect. Well done Grey Cells. Nayama has got it as well, though without reasoning. Nice one Nayama ...

Brhan, my name is NAYANA....yeah. thanks...well, i had forgotten that property of triangles....thanks....i always get lucky with answers...

[Guest]

If you have a 5...

...then the other guy who has a 5 is seeing a 5 and a 7, and sees 3 possible prime totals (17, 19, 23) and can't answer.

...and the other guy who has a 7 is seeing a 5 and a 5, and sees 2 possible prime totals (13 and 17) and can't answer.

If you have a 7...

...then the other guy who has a 5 is seeing a 7 and a 7, and sees 2 possible prime totals (17 and 19) and can't answer.

...then the other guy who has a 7 is seeing a 5 and 7, and sees 3 possible prime totals (17, 19, 23) and can't answer.

If you have an 11...

...then the other guy who has a 5 is seeing a 7 and 11, and sees 3 possible prime totals (23, 29, 31) and can't answer.

...then the other guy who has a 7 is seeing a 5 and 11, and sees 2 possible prime totals (23 and 29) and can't answer.

Some of the choices can be elliminated by applying the triangle inequality -- the sum of two sides is greater than the third side of a triangle.

For instance, if you have a 5 the other guy with 7 sees 5 and 5. So you don't need to consider numbers greater than 9.

[Guest]

I'm either missing something, or there's something missing from the question.

What this answer is saying is that if the person with the 5 sees two 7's, then he knows he has a 5, because 5+7+7 = 19, and 7+7+7=21 (composite) and 11+7+7 = 25 (composite).

But why can't this person think he has a 3? 3+7+7 = 17 (also prime).

While YOU are limited into thinking you have only a 5, 7, or 11 based on the two numbers YOU see, there's no reason that I can see that the other two are limited. There's no reason for THEM to eliminate 3 or 13 as possibilities if they see two numbers on your head and on the remaining person's head that would satisfy the triangle inequality.

Ok, let me elaborate it. There are three possibilities: 5, 7 and 11. Let A is the guy with 5 on his forehead and B the guy with 7 on his forehead.

If you have 5, then B sees (5,5) so he concludes he must have 3 or 7. But 3 can be eliminated because if he (B) has 3 A will see (3,5) and would know his (A) number is 5 since if it is 3 B will see (3,3) making the only solution 5. But the OP states A can't draw a conclusion so you can eliminate 5.

I hope that makes sense. If not write your comments or PM me, and I will elaborate in more depth.

[roolstar]

2nd friend(7) : Sees 5 and my numbers(5,7 or 11).

If he sees 5, his number is 7.

Is 3 a prime?

If yes, then he can be 7 or 3!! (5+5+3=13) and 3 does not violate the triangle property...

And unless 13 is not prime, this would leave the puzzle unsolved then...

No?

Edited April 7, 2008 by roolstar

[Guest]

Is 3 a prime?

If yes, then he can be 7 or 3!! (5+5+3=13) and 3 does not violate the triangle property...

And unless 13 is not prime, this would leave the puzzle unsolved then...

No?

3 is a prime no. but two sides(7,5) are given...u have added 5 two times above.....however thus if u add 3+5+7=15 which is not prime ....hence 3 is ruled out..

[roolstar]

Thank you for your prompt reply Nayana.

But I think you misread my post.

I included below the original post on which I replied. The RED part is where the 3 can fit as a possibility.

Please verify it Brhan.

1st friend(5) : Sees 7 and my number(5,7 or 11).

If he sees 5 , his could be 5,7 or 11.

If he sees 7 , his number is 5

If he sees 11 , his number could be 5,11 or 13.

2nd friend(7) : Sees 5 and my numbers(5,7 or 11).

If he sees 5, his number is 7.

So 5 and 7 are eliminated.

So my number is 11.

Is 3 a prime?

If yes, then he (the 2nd friend with 7 on his forehead who sees 5 and 5) can be 7 or 3!! (5+5+3=13) and 3 does not violate the triangle property...

And unless 13 is not prime, this would leave the puzzle unsolved then...

No?

But I reread the previous posts and Brhan actually demonstrated why 3 cannot be a possibility.

Check POst#21

Edited April 7, 2008 by roolstar

[Guest]

You and your two best friends have numbers written on your foreheads. You are told that the three numbers are primes and that they form the sides of a triangle with prime perimeter. You see 5 and 7 on your friends' forehead, both of whom state that they cannot deduce the number on their own foreheads. What number is written on your forehead?

because for triangle a+b>c, b+c>a, c+a>b

In case of you C

now if a=5, b=7, then c<12....now let us try all the primes below 12..

c=1, in this case a+c<B

c=2, in this case a+c=b

c=3, it follows the rule of triangle, but a+b+c is not prime

c=5, as it follows the rule of triangle and a+b+c=17 is prime

c=7, as it follows the rule of triangle and a+b+c=19 is prime

c=11, as it follows the rule of triangle and a+b+c=23 is prime

So, your numbers can be 5,7, or 11.

What happened to your friend A

he sees b=7, c=5 or 7 or 11,

When c=5, a<12

a=1, in this case a+c<b

a=2, in this case a+c=b

a=3, it follows the rule of triangle, but a+b+c=15 is not prime

a=5, as it follows the rule of triangle and a+b+c=17 is prime

a=7, as it follows the rule of triangle and a+b+c=19 is prime

a=11, as it follows the rule of triangle and a+b+c=23 is prime

So, according to A, his numbers can be 5,7, or 11.

When c=7, a<14

a=1, it follows the rule of triangle, but a+b+c=15 is not prime

a=2, a+b+c=16

a=3, as it follows the rule of triangle and a+b+c=17 is prime

a=5, as it follows the rule of triangle and a+b+c=19 is prime

a=7, a+b+c=21

a=11, a+b+c=25

a=13, a+b+c=27

So, according to A, his numbers can be 1,3, or 5.

When c=11, a<18

a=1, it follows the rule of triangle, but a+b+c=18 is not prime

a=2, as it follows the rule of triangle and a+b+c=19 is prime

a=3, a+b+c=20

a=5, as it follows the rule of triangle and a+b+c=23 is prime

a=7, a+b+c=25

a=11, as it follows the rule of triangle and a+b+c=29 is prime

a=13, as it follows the rule of triangle and a+b+c=31 is prime

a=17, a+b+c=35

So, according to A, his numbers can be 2,5,11,13.

But for all of your options (5,7,11), A's number can be 5.

So, lets think about B

What happened to your friend B

he sees a=5, c=5 or 7 or 11,

When c=5, b<10

b=1, as it follows the rule of triangle and a+b+c=11 is prime

b=2, it follows the rule of triangle , but a+b+c=12

b=3, as it follows the rule of triangle and a+b+c=13 is prime

b=5, a+b+c=15

b=7, as it follows the rule of triangle and a+b+c=17 is prime

So, according to B, his numbers can be 1,3, or 7.

When c=7, b<12

b=1, in this case a+b<c

b=2, in this case a+b=c

b=3, it follows the rule of triangle ,but a+b+c=15

b=5, as it follows the rule of triangle and a+b+c=17 is prime

b=7, as it follows the rule of triangle and a+b+c=19 is prime

b=11, as it follows the rule of triangle and a+b+c=19 is prime

So, according to B, his numbers can be 5,7, or 11.

When c=11, b<16

b=1, as it follows the rule of triangle and a+b+c=17 is prime

b=2, it follows the rule of triangle, but a+b+c=18

b=3, as it follows the rule of triangle and a+b+c=19 is prime

b=5, a+b+c=21

b=7, as it follows the rule of triangle and a+b+c=23 is prime

b=11, a+b+c=26

b=13, as it follows the rule of triangle and a+b+c=29 is prime

So, according to B, his numbers can be 1,3, or 7.

Now, As you can see A's number is 5 and B's number is 7, So, for any of your numbers 5,7 or 11 it is possible….So, the result is undefined.

[Nikyma]

Assuming you can't have a duplicate of the other numbers, it would have to be 11.