Math Problem!!

[smokethedays]

you got 12 Pennies, they ofcourse look identical but one coin is defective weight wise, but we don't know if it's lighter or heavier than the others. given a REGULAR scale (the old style one with two sides) and only 3 trials on the scale (you get to use it only three times) find out the defective coin and determine if it's lighter or heavier. have fun


ps. this problem is from 2005 US national Math Olympiad (national competetion in Math between college students).

[LBT]

Let us denote the coins by C1, C2, C3 etc.. And the pans by P1 and P2.

Put C1, C2, C3 and C4 in P1 and C5, C6, C7 and C8 in the other. We get the following cases.

Case-1: They balance each other.
Now we know that one of C9, C10, C11 and C12 is defective and coins C1 to C8 are non defective.
Remove all the coins. Put C9 and C10 in P1 and C11 and C1 in P2. We get following subcases.

Subcase-1: They balance each other. Now we know that C12 is
defective. Compare it with any other normal coin and find its defect.

Subcase-2: P1 goes down. Now we know that either one of C9 and C10 is heavy or C11 is light. Compare C9 and C10 by putting them in P1 and P2. If P1goes down then C9 is heavy. If P2 goes down then C10 is heavy. If they balance each other then C11 is light.

Subcase-3: P2 goes down. Now we know that either one of C9 and C10 is light or C11 is heavy. Compare C9 and C10 by putting them in P1 and P2. If P1goes up then C9 is light. If P2 goes up then C10 is light. If they balance each other then C11 is heavy.

Case-2: P1 goes down.
Now we know that one of C1, C2, C3 and C4 is heavy or one of C5, C6, C7 and C8 is light and coins C9 to C12 are normal. Let us denote coins C1 to C4 by H1 to H4 and coins C5 to C8 by L1 to L4.

Now put H1, H2, L1 and L2 in P1 and L3, C9, C10 and C11 in P2. Keep H3, H4 and L4 aside. Again we get following subcases.

Subcase-1: They balance each other. Now we know that either one of H3 and H4 is heavy or L4 is light. Compare H3 and H4 by putting them in P1 and P2. If P1 goes down then H3 (C3) is heavy. If P2 goes down then H4 (C4) is heavy. If they balance each other then L4 (C8) is light.

Subcase-2: P1 goes down. Now we know that either one of H1 and H2 is heavy or L3 is light. Compare H1 and H2 by putting them in P1 and P2. If P1 goes down then H1 (C1) is heavy. If P2 goes down then H2 (C2) is heavy. If they balance each other then L3 (C7) is light.

Subcase-3: P2 goes down. Now we know that either L1 or L2 is light. Compare L1 and L2 by putting them in P1 and P2. If P1 goes up then L1 (C5) is light. If P2 goes up then L2 (C6) is light.

Case-3: P2 goes down.
Now we know that Either one of C1, C2, C3 and C4 is light or one of C5, C6, C7 and C8 is heavy. Noe denote coins C1 to C4 by L1 to L4 and coins C5 to C8 by H1 to H4. Now follow the same procedure as in case-2.

[juice_305]

Let us denote the coins by C1, C2, C3 etc.. And the pans by P1 and P2.

Put C1, C2, C3 and C4 in P1 and C5, C6, C7 and C8 in the other. We get the following cases.

Case-1: They balance each other.
Now we know that one of C9, C10, C11 and C12 is defective and coins C1 to C8 are non defective.
Remove all the coins. Put C9 and C10 in P1 and C11 and C1 in P2. We get following subcases.

Subcase-1: They balance each other. Now we know that C12 is
defective. Compare it with any other normal coin and find its defect.

Subcase-2: P1 goes down. Now we know that either one of C9 and C10 is heavy or C11 is light. Compare C9 and C10 by putting them in P1 and P2. If P1goes down then C9 is heavy. If P2 goes down then C10 is heavy. If they balance each other then C11 is light.

Subcase-3: P2 goes down. Now we know that either one of C9 and C10 is light or C11 is heavy. Compare C9 and C10 by putting them in P1 and P2. If P1goes up then C9 is light. If P2 goes up then C10 is light. If they balance each other then C11 is heavy.

Case-2: P1 goes down.
Now we know that one of C1, C2, C3 and C4 is heavy or one of C5, C6, C7 and C8 is light and coins C9 to C12 are normal. Let us denote coins C1 to C4 by H1 to H4 and coins C5 to C8 by L1 to L4.

Now put H1, H2, L1 and L2 in P1 and L3, C9, C10 and C11 in P2. Keep H3, H4 and L4 aside. Again we get following subcases.

Subcase-1: They balance each other. Now we know that either one of H3 and H4 is heavy or L4 is light. Compare H3 and H4 by putting them in P1 and P2. If P1 goes down then H3 (C3) is heavy. If P2 goes down then H4 (C4) is heavy. If they balance each other then L4 (C8) is light.

Subcase-2: P1 goes down. Now we know that either one of H1 and H2 is heavy or L3 is light. Compare H1 and H2 by putting them in P1 and P2. If P1 goes down then H1 (C1) is heavy. If P2 goes down then H2 (C2) is heavy. If they balance each other then L3 (C7) is light.

Subcase-3: P2 goes down. Now we know that either L1 or L2 is light. Compare L1 and L2 by putting them in P1 and P2. If P1 goes up then L1 (C5) is light. If P2 goes up then L2 (C6) is light.

Case-3: P2 goes down.
Now we know that Either one of C1, C2, C3 and C4 is light or one of C5, C6, C7 and C8 is heavy. Noe denote coins C1 to C4 by L1 to L4 and coins C5 to C8 by H1 to H4. Now follow the same procedure as in case-2.

that was going to be my answer

[smokethedays]

Let us denote the coins by C1, C2, C3 etc.. And the pans by P1 and P2.

Put C1, C2, C3 and C4 in P1 and C5, C6, C7 and C8 in the other. We get the following cases.

Case-1: They balance each other.
Now we know that one of C9, C10, C11 and C12 is defective and coins C1 to C8 are non defective.
Remove all the coins. Put C9 and C10 in P1 and C11 and C1 in P2. We get following subcases.

Subcase-1: They balance each other. Now we know that C12 is
defective. Compare it with any other normal coin and find its defect.

Subcase-2: P1 goes down. Now we know that either one of C9 and C10 is heavy or C11 is light. Compare C9 and C10 by putting them in P1 and P2. If P1goes down then C9 is heavy. If P2 goes down then C10 is heavy. If they balance each other then C11 is light.

Subcase-3: P2 goes down. Now we know that either one of C9 and C10 is light or C11 is heavy. Compare C9 and C10 by putting them in P1 and P2. If P1goes up then C9 is light. If P2 goes up then C10 is light. If they balance each other then C11 is heavy.

Case-2: P1 goes down.
Now we know that one of C1, C2, C3 and C4 is heavy or one of C5, C6, C7 and C8 is light and coins C9 to C12 are normal. Let us denote coins C1 to C4 by H1 to H4 and coins C5 to C8 by L1 to L4.

Now put H1, H2, L1 and L2 in P1 and L3, C9, C10 and C11 in P2. Keep H3, H4 and L4 aside. Again we get following subcases.

Subcase-1: They balance each other. Now we know that either one of H3 and H4 is heavy or L4 is light. Compare H3 and H4 by putting them in P1 and P2. If P1 goes down then H3 (C3) is heavy. If P2 goes down then H4 (C4) is heavy. If they balance each other then L4 (C8) is light.

Subcase-2: P1 goes down. Now we know that either one of H1 and H2 is heavy or L3 is light. Compare H1 and H2 by putting them in P1 and P2. If P1 goes down then H1 (C1) is heavy. If P2 goes down then H2 (C2) is heavy. If they balance each other then L3 (C7) is light.

Subcase-3: P2 goes down. Now we know that either L1 or L2 is light. Compare L1 and L2 by putting them in P1 and P2. If P1 goes up then L1 (C5) is light. If P2 goes up then L2 (C6) is light.

Case-3: P2 goes down.
Now we know that Either one of C1, C2, C3 and C4 is light or one of C5, C6, C7 and C8 is heavy. Noe denote coins C1 to C4 by L1 to L4 and coins C5 to C8 by H1 to H4. Now follow the same procedure as in case-2.

sorry man, but misread. yeah you got it

[UpstateTank]

my head hurts from reading all of that

[Phreak101]

Bring back the political forum

[pepperoni]

The only thing i don't get is when you put two pennies onto the scale on opposite sides and one side goes down which would mean the other side would go up right? Is it a counter-balance scale, so as one side drops the other raises. So if this happens how would u know if one side has a heavier than normal penny or the other side has a lighter than normal penny. Because one side could drop so you would assume that it is heavier.... but it could be normal and just weigh more than the other side because that is the defective lighter penny. I don't get it

[juice_305]

i hate math i just stick to weights!!!!!!

[smokethedays]

i hate math i just stick to weights!!!!!!

u need math to know the weights

[Doc.Sust]

????dont know where to even start

[Doc.Sust]

i 2nd that, i hate math as well, but it is a necessary evil for technology and science.