Interesting Math Problems

[rule34]

220, 240, whatever it takes.

[QillerDaemon]

this is like wordle but with numbers.

Ain't that the truth!

I didn't answer, as I've seen variants of this problem before. In fact, a logic professor used a variant as an "extra" credit question. We could either do the test as is, or just answer a similar problem instead, either way for full credit. Almost everyone (including myself) just did the test as is.

[Antknot]

the solution looks much more complicated than it is.

Basically, for each wrap the string has to make a hypotenuse of a right triangle formed by the Circumference and the Pitch (Length/# Turns).

In this case that is the hypotenuse of a right triangle with leg 4 and 3. That hypotenuse is 5. Now how many times does it do that? In this case it makes 4 turns. so that is 4 x 5 = 20.

It was not specified the start and stop points were the ends of the rod.

My 16 is correct. My 48 is incorrect it should be 96.

[Animal]

A string is wrapped around a rod exactly four times end to end. The rod is 12 units long and has a circumference of 4 units. What is the length of the string?

Edit: the string wraps around the rod exactly like the red strip around the barber's pole, screw thread like. Just so we're clear, since I haven't included a picture.

By the way, wrong answers so far. Hint: split the rod and thread length-wise and unfold.

[WarmBidetWater]

the solution looks much more complicated than it is.

Basically, for each wrap the string has to make a hypotenuse of a right triangle formed by the Circumference and the Pitch (Length/# Turns).

In this case that is the hypotenuse of a right triangle with leg 4 and 3. That hypotenuse is 5. Now how many times does it do that? In this case it makes 4 turns. so that is 4 x 5 = 20.

It was not specified the start and stop points were the ends of the rod.

My 16 is correct. My 48 is incorrect it should be 96.

I agree. The problem description was highly flawed. Which is why I typically don't participate in these sort of posts.

[Animal]

394

winner, winner! chicken dinner!

[QillerDaemon]

A string is wrapped around a rod exactly four times end to end. The rod is 12 units long and has a circumference of 4 units. What is the length of the string?

Edit: the string wraps around the rod exactly like the red strip around the barber's pole, screw thread like. Just so we're clear, since I haven't included a picture.

By the way, wrong answers so far. Hint: split the rod and thread length-wise and unfold.

In all fairness, that was an edit when I realized some were conceptualizing the problem incorrectly. So I can see where the answers went awry.

[Reservoir Dog]

The answer is, and always will be, 42.

[Bluespruce1964]

20

[Bluespruce1964]

[Hyperv12]

165?

[QillerDaemon]

A to power B = 343, B to power C = 10, A to power C = 7. What is B to power B?
(edit: hint, the values of A, B, and C really don't matter, other than they don't equal each other)

[Animal]

A to power B = 343, B to power C = 10, A to power C = 7. What is B to power B?
(edit: hint, the values of A, B, and C really don't matter, other than they don't equal each other)

10 to the 10th power. Or 10,000,000,000

[QillerDaemon]

A to power B = 343, B to power C = 10, A to power C = 7. What is B to power B?
(edit: hint, the values of A, B, and C really don't matter, other than they don't equal each other)

10 to the 10th power. Or 10,000,000,000

Buzz! You're way out of degree. Think about what the number 343 is.

[rule34]

A to power B = 343, B to power C = 10, A to power C = 7. What is B to power B?
(edit: hint, the values of A, B, and C really don't matter, other than they don't equal each other)

What kind of power? Gas, electric, nuclear?

[Animal]

A to power B = 343, B to power C = 10, A to power C = 7. What is B to power B?
(edit: hint, the values of A, B, and C really don't matter, other than they don't equal each other)

10 to the 10th power. Or 10,000,000,000

Buzz! You're way out of degree. Think about what the number 343 is.

Yeah, I just looked at it again and I wasn't thinking.

A^B must be 7^3. So, A is 7 and B is 3.

A^C must be 7^1. So, A is 7 and C is 1.

B^C must be B^1 = 10. So B = 10.

Now, I guess that means we have to do B^B = 10^3? or 1,000? I was confused that there are two different numbers for B.

It would make more sense to use Capital B and Lower case B.

[QillerDaemon]

Yeah, I just looked at it again and I wasn't thinking.

A^B must be 7^3. So, A is 7 and B is 3.

A^C must be 7^1. So, A is 7 and C is 1.

B^C must be B^1 = 10. So B = 10.

Now, I guess that means we have to do B^B = 10^3? or 1,000? I was confused that there are two different numbers for B.

It would make more sense to use Capital B and Lower case B.

You have the right numerical answer, but you're still worrying about the actual values of A, B, and C when you don't need to. What their values are don't matter, they can be anything using the exponential change of base rule (except 1). Try again, but consider the exponential power rule instead. That way, again the variables can have any actual value, not just what you figured them to be. You'll still come to the same answer.

[Animal]

Yeah, I just looked at it again and I wasn't thinking.

A^B must be 7^3. So, A is 7 and B is 3.

A^C must be 7^1. So, A is 7 and C is 1.

B^C must be B^1 = 10. So B = 10.

Now, I guess that means we have to do B^B = 10^3? or 1,000? I was confused that there are two different numbers for B.

It would make more sense to use Capital B and Lower case B.

You have the right numerical answer, but you're still worrying about the actual values of A, B, and C when you don't need to. What their values are don't matter, they can be anything using the exponential change of base rule (except 1). Try again, but consider the exponential power rule instead. That way, again the variables can have any actual value, not just what you figured them to be. You'll still come to the same answer.

Well, I don't follow that, but it seems to me that in your question B means one thing if it is being raised to a power and B means something different if it represents a power that it is raising something else to. So B^B = 10^3 or 1,000.

I could argue that B^B might also equal 3^10 or 59,049 but I wouldn't guess that answer since that means switching the B's in the way they are used in the question. Same reason I rule out 10^10 (my first guess) or 3^3 for an answer.

[Antknot]

Yeah, I just looked at it again and I wasn't thinking.

A^B must be 7^3. So, A is 7 and B is 3.

A^C must be 7^1. So, A is 7 and C is 1.

B^C must be B^1 = 10. So B = 10.

Now, I guess that means we have to do B^B = 10^3? or 1,000? I was confused that there are two different numbers for B.

It would make more sense to use Capital B and Lower case B.

You have the right numerical answer, but you're still worrying about the actual values of A, B, and C when you don't need to. What their values are don't matter, they can be anything using the exponential change of base rule (except 1). Try again, but consider the exponential power rule instead. That way, again the variables can have any actual value, not just what you figured them to be. You'll still come to the same answer.

J.A.F.U.Q.M.Q.

[stonedmegman]

Yeah, I just looked at it again and I wasn't thinking.

A^B must be 7^3. So, A is 7 and B is 3.

A^C must be 7^1. So, A is 7 and C is 1.

B^C must be B^1 = 10. So B = 10.

Now, I guess that means we have to do B^B = 10^3? or 1,000? I was confused that there are two different numbers for B.

It would make more sense to use Capital B and Lower case B.

You have the right numerical answer, but you're still worrying about the actual values of A, B, and C when you don't need to. What their values are don't matter, they can be anything using the exponential change of base rule (except 1). Try again, but consider the exponential power rule instead. That way, again the variables can have any actual value, not just what you figured them to be. You'll still come to the same answer.

J.A.F.U.Q.M.Q.

You have to divide by E.I.E.I.O.

[QillerDaemon]

Well, I don't follow that, but it seems to me that in your question B means one thing if it is being raised to a power and B means something different if it represents a power that it is raising something else to. So B^B = 10^3 or 1,000.

I could argue that B^B might also equal 3^10 or 59,049 but I wouldn't guess that answer since that means switching the B's in the way they are used in the question. Same reason I rule out 10^10 (my first guess) or 3^3 for an answer.

What I meant was that the actual number values of the three variables are A, B, and C. Nothing more or less. Whether the variable is being used as the base or the exponent, its value is exactly the same and meaningless. As my old math professor used to say "you got the right answer, but you didn't use the information provided in the correct manner." This problem is strictly an algebraic question using the power rule and the info provided, *not* to figure out what the value of A, B, and C each is.

Note: I'm changing the notation a little. I wish we had sub- and superscripts in phpBB, we don't. So instead of saying "A to power B", I'm changing it to "A power B", with "power" signifying the exponential power. Maybe that clarifies things?

The original question is "A power B = 343, B power C = 10, A power C = 7. What is B power B?" In all actuality, the values of A, B, and C can be anything, and ultimately what the value of B is, isn't of any concern, it's what the value of B power B is that matters.

343 is 7 cubed, 7 power 3. Now A power B = 343 = 7 power 3. We do not necessarily know that A = 7 or that B = 3 at this point.

Substitute into that A power C = 7 ("using the info provided"), and you get A power B = (A power C) power 3. Use the power rule to make A power B = A power 3C. Now you have an equation with the same base (whatever that value of the base is), so the two powers have to be equal.

So B = 3C. Go back to B power B. Substitute in 3C for the exponent B: B power B = B power 3C. Use the power rule again, using its inverse, to get B power B = (B power C) power 3. The second given equation is B power C = 10, substitute that into the parentheses.

B power B = (10) power 3. At no time do we know the actual values of A, B, or C, nor care.

[Animal]

Well, I don't follow that, but it seems to me that in your question B means one thing if it is being raised to a power and B means something different if it represents a power that it is raising something else to. So B^B = 10^3 or 1,000.

I could argue that B^B might also equal 3^10 or 59,049 but I wouldn't guess that answer since that means switching the B's in the way they are used in the question. Same reason I rule out 10^10 (my first guess) or 3^3 for an answer.

What I meant was that the actual number values of the three variables are A, B, and C. Nothing more or less. Whether the variable is being used as the base or the exponent, its value is exactly the same and meaningless. As my old math professor used to say "you got the right answer, but you didn't use the information provided in the correct manner." This problem is strictly an algebraic question using the power rule and the info provided, *not* to figure out what the value of A, B, and C each is.

Note: I'm changing the notation a little. I wish we had sub- and superscripts in phpBB, we don't. So instead of saying "A to power B", I'm changing it to "A power B", with "power" signifying the exponential power. Maybe that clarifies things?

The original question is "A power B = 343, B power C = 10, A power C = 7. What is B power B?" In all actuality, the values of A, B, and C can be anything, and ultimately what the value of B is, isn't of any concern, it's what the value of B power B is that matters.

343 is 7 cubed, 7 power 3. Now A power B = 343 = 7 power 3. We do not necessarily know that A = 7 or that B = 3 at this point.

Substitute into that A power C = 7 ("using the info provided"), and you get A power B = (A power C) power 3. Use the power rule to make A power B = A power 3C. Now you have an equation with the same base (whatever that value of the base is), so the two powers have to be equal.

So B = 3C. Go back to B power B. Substitute in 3C for the exponent B: B power B = B power 3C. Use the power rule again, using its inverse, to get B power B = (B power C) power 3. The second given equation is B power C = 10, substitute that into the parentheses.

B power B = (10) power 3. At no time do we know the actual values of A, B, or C, nor care.

you actually do care what A,B, and C are because you have to remember to keep them in the right places as they relate to the equation. I follow what you are doing, but its just a much more complicated way of doing what I did.

A^B = 343. That only has one answer. It must be 7^3 = 343. So A=7 and B=3 (when its used as a power).

A^C = 7. Since we know A=7, 7^C=7. So, C=1.

Now, B^C = 10. Since we don't know what B is when its used as a Base number we have to plug in C. B^1 = 10. So B=10 when used as a base.

Now B^B = 10^3 = 1,000.

[Animal]

But, the question should be worded:

A^b = 343
B^c = 10
A^c = 7

What is B^b?

[QillerDaemon]

A_B = B^A

_Z -- ^X -- ^{whole lotta text above the line}

A_BCD -- D^CBA

A_B^CDEFG

wasn't that hard... Just have to keep the tags straight.

A^B = 343, B^C = 10, A^C = 7. What is B^B?

[QillerDaemon]

But, the question should be worded:
logrithmic
A^b = 343
B^c = 10
A^c = 7

What is B^b?

No, and here's why: this is a question from a national mathematics olympiad. These questions are designed around two strict ideas, one that you come to the answer as quickly as possible, and two that you use the problem and given facts about the problem as efficiently and fully as possible in accordance with using best math solving strategies. You can still have the right answer and be marked wrong due to how badly the judges see you use the given info or where your strategy forks. These kinds of questions do not have extraneous information, and the information is meant to be used completely towards answering the problem. You are not supposed to re-think or question the information, you don't have time for that shit.

Going back to the original question, you were given the question of what B^B is based on the three given equalities. You can solve for the answer without bothering to try to answer what the exact values of A, B, and C are. You don't need to worry about their values, because you can use the given equalities to do that. Can you tell me what number B is that B multiplied by itself B times would make 1000? Equivalently, can you tell me what number B is that [B x log B = 3]? No, but using the given equalities gets you around that.

You said that A^B = 343 = 7³ means A = 7 and B = 3. That could be true, but it could also be the logarithmic equivalent of dividing by a hidden zero. The valid step is to make the bases both be A on either side of the equation. That way you don't need to know what the value of A ever is directly. But since they are the same base A, their exponents are equal. That's valid, and we never need to concern ourselves with whether A really is 7 or not.