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Counterfeit Treasure A

Hunting for treasure at the end of a rainbow one day you come across a Leprechaun. He tells you that he will share a portion of his treasure with you if you can pass a test. You agree and the Leprechaun produces a golden two pan balance scale and 15 coins out of thin air.

"One of the coins before you is fake! You will know the fake coin because it's weight is very slightly different than the others. You may use the balance, but no more than 3 times. Find me the fake and tell me whether it is too light or too heavy and the treasure is yours! I'll even help you out a bit."

The Leprechaun pulls one of the 15 coins aside and says "This coin is real."

He pulls aside another coin and says "If this coin is the fake then it is lighter than the others."

Describe a method you can use to guarantee finding the fake (and whether it is lighter or heavier) using the scale no more than 3 times.

Counterfeit Treasure B

Hunting for treasure at the end of a rainbow one day you come across a Leprechaun. He tells you that he will share a portion of his treasure with you if you can pass a test. You agree and the Leprechaun produces a golden two pan balance scale and 4 coins out of thin air.

"Two of these coins are fake! You will know a fake coin because it's weight will be either a quarter ounce heavier or a quarter ounce lighter than a real coin. You may use the balance, but no more than 3 times. Find me both fakes and tell me whether each is too light or too heavy and the treasure is yours! I'll even help you out a bit."

The Leprechaun produces a new handful of coins and says "These coins are all real. You can use them if you think it will help you, but I want them back if you fail!"

Describe a method you can use to guarantee finding the fakes (and whether each is lighter or heavier) using the scale no more than 3 times. (Note: It is possible for one of the fake coins to be lighter while the other is heavier)

-Tuck

The 15 coins are 1-9,A,B,C,D,S and R. Ris the real coin and S is the special coin that, if it's fake, islight.

Format:

Result of weighing 1 : Result of 2 : Result of 3 -> Fake coin and Weight (L for light, H for heavy).

= means the scales balance

> means the left side is heavy

< means the right side is heavy

12345=6789R : ABC=123 : D=1 -> SL

12345=6789R : ABC=123 : D>1 -> DH

12345=6789R : ABC=123 : D<1 -> DL

12345=6789R : ABC>123 : A=B -> CH

12345=6789R : ABC>123 : A>B -> AH

12345=6789R : ABC>123 : A<B -> BH

12345=6789R : ABC<123 : A=B -> CL

12345=6789R : ABC<123 : A>B -> BL

12345=6789R : ABC<123 : A<B -> AL

12345>6789R : 3459=12AB : 6=7 -> 8L

12345>6789R : 3459=12AB : 6>7 -> 7L

12345>6789R : 3459=12AB : 6<7 -> 6L

12345>6789R : 3459>12AB : 3=4 -> 5H

12345>6789R : 3459>12AB : 3>4 -> 3H

12345>6789R : 3459>12AB : 3<4 -> 4H

12345>6789R : 3459<12AB : 1=2 -> 9L

12345>6789R : 3459<12AB : 1>2 -> 1H

12345>6789R : 3459<12AB : 1<2 -> 2H

12345<6789R : 3459=12AB : 6=7 -> 8H

12345<6789R : 3459=12AB : 6>7 -> 7H

12345<6789R : 3459=12AB : 6<7 -> 6H

12345<6789R : 3459>12AB : 1=2 -> 9H

12345<6789R : 3459>12AB : 1>2 -> 2L

12345<6789R : 3459>12AB : 1<2 -> 1L

12345<6789R : 3459<12AB : 3=4 -> 5L

12345<6789R : 3459<12AB : 3>4 -> 4L

12345<6789R : 3459<12AB : 3<4 -> 3L

The 4 unknown coins are 1,2,3 and 4. R is one of the real coins.

Format: Result of weighing 1 : Result of 2 : Result of 3 -> Fake coin and Weight (L for light, H for heavy).

= means the scales balance

> means the left side is heavy

< means the right side is heavy

1=2 : 13=2R : 1>R : 1H2H

1=2 : 13=2R : 1<R : 1L2L

1=2 : 13>2R : 4>R : 3H4H

1=2 : 13>2R : 4<R : 3H4L

1=2 : 13<2R : 4>R : 3L4H

1=2 : 13<2R : 4<R : 3L4L

1>2 : 12=3R : 3=R : 1H2L

1>2 : 12=3R : 3>R : 1H3H

1>2 : 12=3R : 3<R : 2L3L

1>2 : 12>3R : 4=R : 1H3L

1>2 : 12>3R : 4>R : 1H4H

1>2 : 12>3R : 4<R : 1H4L

1>2 : 12<3R : 4=R : 2L3H

1>2 : 12<3R : 4>R : 2L4H

1>2 : 12<3R : 4<R : 2L4L

1<2 : 12=3R : 3=R : 1L2H

1<2 : 12=3R : 3>R : 2H3H

1<2 : 12=3R : 3<R : 1L3L

1<2 : 12>3R : 4=R : 2H3L

1<2 : 12>3R : 4>R : 2H4H

1<2 : 12>3R : 4<R : 2H4L

1<2 : 12<3R : 4=R : 1L3H

1<2 : 12<3R : 4>R : 1L4H

1<2 : 12<3R : 4<R : 1L4L

You could also begin with 12 vs RR or 123 VS RRR

Edited by bonanova
Replace solutions at owner request
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plainglazed Grand Master

plainglazed

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Label the coins 1-15 with 1 being known to be real {R} and 15 to be light if not real:

1) Compare 1{R},2,3,4,5 and 6,7,8,9,10

if R,2,3,4,5 > 6,7,8,9,10 then one of either 2,3,4, or 5 is heavy or one of 6,7,8,9,10 is light and 11,12,13,14,15 are real {R}.

2) Compare 2,3,4,6,7,8 and R,R,R,R,R,R

if 2,3,4,6,7,8 > R,R,R,R,R,R then either 2,3, or 4 is heavy

3) Compare 2 and 3

if 2>3 then 2 is heavy, if 2<3 than 3 is heavy, if 2=3 then 4 is heavy

if 2,3,4,6,7,8 < R,R,R,R,R,R then either 6,7, or 8 is light

3) Compare 6 and 7

if 6>7 then 7 is light, if 6<7 then 6 is light, if 6=7 then 8 is light

if 2,3,4,6,7,8 = R,R,R,R,R,R then either 5 is heavy or 9 or 10 is light

3) Compare 9 and 10

if 9>10 then 10 is light, if 9<10 then 9 is light, if 9=10 then 5 is heavy

if R,2,3,4,5 < 6,7,8,9,10 the same methodology as above applies.

if R,2,3,4,5 = 6,7,8,9,10 then 11,12,13,14 are either heavy or light or 15 is light

2) Compare 11,12,13, and R,R,R

if 11,12,13 > R,R,R then either 11,12, or 13 are heavy

3) Compare 11 and 12

if 11>12 then 11 is heavy, if 11<12 then 12 is heavy, if 11=12 then 13 is heavy

if 11,12,13 < R,R,R the same methodology as above applies

if 11,12,13 = R,R,R then either 14 is heavy or light or 15 is light

3) Compare 14 and R

if 14>R then 14 is heavy, if 14<R then 14 is light, if 14=R then 15 is light.

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plainglazed Grand Master

plainglazed

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Label the coins 1,2,3,4 and the "real" coins R:

1) Compare 1,2,3 and R,R,R

if 1,2,3 = R,R,R then there must be one light and one heavy coin in either 1,2,3

2) Compare 1 and 2

if 1>2 then either 1H,2L or 1H,3L or 3H,2L

3) Compare 3 and R

if 3>R then 3H,2L or if 3<R then 1H,3L or if 3=R then 1H,2L

if 1<2 then either 1L,2H or 1L,3H or 2H,3L

3) Compare 3 and R as above

if 1,2,3 > R,R,R then either 1H,2H; 1H,3H; 2H,3H; 1H,4H; 1H,4L; 2H,4H; 2H,4L; 3H,4H; 3H,4L

2) Compare 1 and 2

if 1>2 then either 1H,3H or 1H,4H or 1H,4L

3) Compare 4 and R

if 4>R then 1H,4H or if 4<R then 1H,4L or if 4=R then 1H,3H

if 1<2 then either 2H,3H or 2H,4H or 2H,4L

3) Compare 4 and R

if 4>R then 2H,4H or if 4<R then 2H,4L or if 4=R then 2H,3H

if 1=2 then either 1H,2H or 3H,4H or 3H,4L

3) Compare 4 and R

if 4>R then 3H,4H or if 4<R then 3H,4L or if 4=R then 1H,2H

if 1,2,3 < R,R,R then either 1L,2L; 1L,3L; 2L,3L; 1L,4H; 1L,4L; 2L,4H; 2L,4L; 3L,4H; 3L,4L

2) Compare 1 and 2 then 3) Compare 4 and R as above

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Guest

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Ding and Ding!

Nice work Plainglazed!

Here's my 'official' solutions. Not sure how I get them put onto the OP again :P

The 15 coins are 1-9,A,B,C,D,S and R. R is the real coin and S is the special coin that, if it's fake, is light.

Format:

Result of weighing 1 : Result of 2 : Result of 3 -> Fake coin and Weight (L for light, H for heavy).

= means the scales balance

> means the left side is heavy

< means the right side is heavy

12345=6789R : ABC=123 : D=1 -> SL

12345=6789R : ABC=123 : D>1 -> DH

12345=6789R : ABC=123 : D<1 -> DL

12345=6789R : ABC>123 : A=B -> CH

12345=6789R : ABC>123 : A>B -> AH

12345=6789R : ABC>123 : A<B -> BH

12345=6789R : ABC<123 : A=B -> CL

12345=6789R : ABC<123 : A>B -> BL

12345=6789R : ABC<123 : A<B -> AL

12345>6789R : 3459=12AB : 6=7 -> 8L

12345>6789R : 3459=12AB : 6>7 -> 7L

12345>6789R : 3459=12AB : 6<7 -> 6L

12345>6789R : 3459>12AB : 3=4 -> 5H

12345>6789R : 3459>12AB : 3>4 -> 3H

12345>6789R : 3459>12AB : 3<4 -> 4H

12345>6789R : 3459<12AB : 1=2 -> 9L

12345>6789R : 3459<12AB : 1>2 -> 1H

12345>6789R : 3459<12AB : 1<2 -> 2H

12345<6789R : 3459=12AB : 6=7 -> 8H

12345<6789R : 3459=12AB : 6>7 -> 7H

12345<6789R : 3459=12AB : 6<7 -> 6H

12345<6789R : 3459>12AB : 1=2 -> 9H

12345<6789R : 3459>12AB : 1>2 -> 2L

12345<6789R : 3459>12AB : 1<2 -> 1L

12345<6789R : 3459<12AB : 3=4 -> 5L

12345<6789R : 3459<12AB : 3>4 -> 4L

12345<6789R : 3459<12AB : 3<4 -> 3L

The 4 unknown coins are 1,2,3 and 4. R is one of the real coins.

Format: Result of weighing 1 : Result of 2 : Result of 3 -> Fake coin and Weight (L for light, H for heavy).

= means the scales balance

> means the left side is heavy

< means the right side is heavy

1=2 : 13=2R : 1>R : 1H2H

1=2 : 13=2R : 1<R : 1L2L

1=2 : 13>2R : 4>R : 3H4H

1=2 : 13>2R : 4<R : 3H4L

1=2 : 13<2R : 4>R : 3L4H

1=2 : 13<2R : 4<R : 3L4L

1>2 : 12=3R : 3=R : 1H2L

1>2 : 12=3R : 3>R : 1H3H

1>2 : 12=3R : 3<R : 2L3L

1>2 : 12>3R : 4=R : 1H3L

1>2 : 12>3R : 4>R : 1H4H

1>2 : 12>3R : 4<R : 1H4L

1>2 : 12<3R : 4=R : 2L3H

1>2 : 12<3R : 4>R : 2L4H

1>2 : 12<3R : 4<R : 2L4L

1<2 : 12=3R : 3=R : 1L2H

1<2 : 12=3R : 3>R : 2H3H

1<2 : 12=3R : 3<R : 1L3L

1<2 : 12>3R : 4=R : 2H3L

1<2 : 12>3R : 4>R : 2H4H

1<2 : 12>3R : 4<R : 2H4L

1<2 : 12<3R : 4=R : 1L3H

1<2 : 12<3R : 4>R : 1L4H

1<2 : 12<3R : 4<R : 1L4L

You could also begin with 12 vs RR or 123 VS RRR

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Guest

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let the coins be A,B,C,D,E,F,G,H,I,J,K,L,M,N,O

-- let J be the coin we know is real and let O be the coin which can only be real or light (sorry if this is awkward, but I was solving this as typing it up)

1) Weigh ABCDE vs FGHIJ

if ABCDE is heavier

(ABCDE is heavy or FGHIJ is light)

2) Weigh ABF vs. DEG

if ABF is heavier

(AB is heavy or G is light)

3) Weigh A vs. B

if A is heavier, A is heavy

if B is heavier, B is heavy

if equal, G is light

if DEG is heavier

(DE is heavy or F is light)

3) Weigh D vs. E

if D is heavier, D is heavy

if E is heavier, E is heavy

if equal, F is light

if equal

(C is heavy or HIJ is light)

3) Weigh H vs. I

if H is heavier, I is light

if I is heavier, H is light

if equal, C is heavy

if FGHIJ is heavier

(ABCDE is light or FGHIJ is heavy)

2) Weigh ABF vs. DEG

if ABF is lighter

(AB is light or G is heavy)

3) Weigh A vs. B

if A is heavier, B is light

if B is heavier, A is light

if equal, G is heavy

if DEG is lighter

(DE is light or F is heavy)

3) Weigh D vs. E

if D is heavier, E is light

if E is heavier, D is light

if equal, F is heavy

if equal

(C is light or HIJ is heavy)

3) Weigh H vs. I

if H is heavier, H is heavy

if I is heavier, I is heavy

if equal, C is light

if equal

2) Weigh AKL vs. BCM

if AKL is heavier

(KL is heavy or M is light)

3) Weigh K vs. L

if K is heavier, K is heavy

if L is heavier, L is heavy

if equal, M is light

if AKL is lighter

(KL is light or M is heavy)

3) Weigh K vs. L

if K is heavier, L is light

if L is heavier, K is light

if equal, M is heavy

if equal

(N or O is fake)

3) Weigh A vs. N

if A is heavier, N is light

if N is heavier, N is heavy

if equal, O is fake, and therefore light.

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