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(13/06) Warbringer : I am really sorry for this, but I forgot something in the answer I sent previously, and I thought I'd correct that. Although it looks obvious, I just wanted to make sure that everything is perfectly clear.
In Case 1.1, I treated the case where 1,5,6 and 2,7,8 don't balance, and 1,5,6 is heavier. I forgot about the case where 2,7,8 is heavier. It's actually the same logic : the fake shuriken would be 7, 8, or 1. If it is either 7 or 8, it would be heavier than the others, else it would be lighter. We weight 7 and 8 (Move 4). If they balance, the fake shuriken is 1, else it is the heavier of 7 and 8.
Shika: I get what you mean, you're right. This is getting really surreal as once again, Cecil beat you to it. There is however a simpler way also involving piles of 6, 6 & 8. I'll reveal the solution soon after a nap.
(14/06) Doink : Man, this one is easier than the first!! or maybe i didnt think of the first one hard enough! anyway, here goes... Divide the 20 shurikens into 4 piles, 3 piles of 6 and 1 pile of 2. That means we have groups A, B and C with 6 shurikens and group D with 2 shurikens. This question can be solved in many ways, depends on how the weight balances out. I will go through all possibilities.
Solution 1. First weigh A with B. Say it balances out. Then weigh either A or B with C. Say it balances out too. Then you will be left with D, and you know the fake shuriken is in D. Take 2 shurikens out from either A, B or C, and weigh it with D. You will now determine whether the fake shuriken is lighter or heavier (Heavier if the D pile is heavier and vice versa). Weigh the last two shurikens in D and voila! Problem solved.
Solution 2. Weigh A and B. Say it balances out. Then weight either A or B with C. Doesnt balance out. If C was heavier we know that the fake shuriken inside C is heavier and vice versa. Break the C pile into groups E and F, E having 4 shurikens and F 2 shurikens. Weigh pile E with itself ( 2 - 2 ). If it balances out den the fakey is in pile F. If not its in pile E. Since we already know that whether the fake shuriken is heavier or lighter, we can easily determine where it is. Just say we found out the fakey is heavier. If the E pile was unbalanced we take the heavier pile, and do the last weigh! If it was pile F, well.. i guess everything reveals itself... =)
Solution 3. Weigh A and B. Doesnt balance out. Note which is heavier and which is lighter. Weight either one with C. Lets say its A with C. Balances out. So the fakey is in pile B. If B was the heavier pile earlier then the fakey is heavier and vice versa. If A and C doesnt balance out then the fakey is in pile A. Again, whether the fakey is heavier or lighter is determined as above. We have 2 more times to use the balancing scales. Like in Solution 2, break the fakey pile into 4's and 2's, and repeat the above! =)
Shika : Now that's a more comprehensive answer! I'll use this as our model answer. 3rd one to get it right!
(14/06) Reverend : Split the 20 shurikens into piles of 6, with 2 remaining. Naming the piles of 6 as A B and C, and the pile of 2 as D.
Leaving aside D first, I'll balance with A B and C.
1st move. Balance A and B. If A and B are balanced, it means that the fake shuriken is in pile C or D. If A and B are not balanced, and A dips/lifts, then measure A and C. If A and C are balanced, then the fake shuriken is in pile B. If A and C are not balanced, and A dips/lifts, then the fake shuriken is in pile A, vice versa.
2nd move. Balance A and C. If A and C are balanced, it means that the fake shuriken is in pile D. If A and B are not balanced, and A dips/lifts, the fake shuriken is in pile C, and vice versa.
During the 1st and 2nd moves, the pile in which the fake shuriken is in will either dip or lift. If it dips, it means that the fake shuriken is heavier. If the pile lifts, it means that the fake shuriken is lighter.
3rd move. Split the 6 shurikens into 2 piles of 3. Balance them on the scales. then you can tell which pile the fake shuriken is in because you already know if it is heavier or lighter.
4th move. Put aside 1 of the shuriken where the fake shuriken is, balance the remaining 2 on the scale. If the scale is balanced, it means that the fake shuriken is the one you left aside. If the scale is unbalanced, then the one that is heavier/lighter is the fake one, which you already deduced from the first 2 moves.
If after the 2nd move, piles A B and C are all weights, then the fake shuriken is in pile D. Take one real shuriken as a measure, and balance the 2 shurikens against the real one. The fake shuriken is revealed!
Shika : Another correct one! Think I should start setting more difficult questions...
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