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Shikamaru's Maths Challange

*This challenge was put up on 10 June 2004 and solved on 14 June 2004 with 10 answers, 4 of which are correct, and 6 wrong ones.

Challenge #02 - Find the Fake Shuriken

 
  Question: You're at the weapons shop buying new shurikens as you used all of them in the last annoying mission that involved scrolls and explosive tags... You asked for 20 shurikens but the dishonest shopkeeper gave you a fake one hidden in the 19 real ones. The fake shuriken looks and feels exactly the same as the real ones, except for a slight weight difference. You've been warned by the jounins about this shopkeeper and they told you a way to find out which shuriken is fake by using a balancing scale. However, you do not know whether the fake shuriken is heavier or lighter than the real ones.

Find the fake shuriken in 4 moves using the scales (meaning you're only allowed to place the shurikens on the balancing scale 4 times), and not hurting the shopkeeper with it.

Click here to read the answers of the very brave Shinobis who took up the challenge.

Shikamaru's Answer: This is a question so much easier than the 1st. 4 shinobis got it right. 1 drew diagrams, 1 made my eyes hurt reading the extensive text. 1 gave short and sweet answers. I'll use Doink's submission as a model answer. You can read the rest of the correct answers here.

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! =)

 



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