Zeus's Master bolt

Number Theory Level 3

Zeus is the greek god of the skies. His lightening staff( Master bolt has been stolen. He suspects and infact he is quite sure that it has been stolen by someone from camp Halfblood ( a camp for demigods). So he called all the camp Halfblood members. Zeus then commanded them to throw their bagpacks infront of him so that he can check them. He knows that the masterbolt is quite heavy, so it will make the backpack in which it is contained quite heavier from the other. However there are 270 members in the camp ( therefore 270 bags to check) . He summoned a huge balance scale to weigh the all the bags and find the bag containing the Master bolt. However, Athena , the goddess of wisdom approached him and suggested him a way to use the scale as least as possible and find the bag. She told him the minimum number of times he needs to use the scale and how. How many times did she suggest him?

maximum 5 , minimum 4 135 ( if not found earlier by chance) minimum 134, maximum 135 14

3 solutions

Daniel Liu
Feb 28, 2014

A general formula: Let the number of bags be $k$ . Find the integer $n$ that satisfies $3^{n-1}<k\le 3^{n}$ ; the minimum number times needed to use the balance is $n$ .

In this case, $3^5<270< 3^6$ , so the minimum numbers of time Zeus needed to use the scale was $\boxed{6}$ .

There are no right answer choices, so I just picked the closest one.

D K
Feb 23, 2014

There are 270 bags. So, divide them into 3 groups of 90 bags. 1-->Now, weigh 2 of the three groups. If one of the 2 groups is heavier, it has the bolt. If both of them have same weight , the bolt is in the 3rd group. 2--> Now divide the group of 30 we got into 3 groups of 10. By the similar method as above we will get the group having the bolt. 3--> Now divide the group of 10 we got into 3 groups of 3 and 1 is left behind. Weigh 2 of the groups of 3.If one appears to have a larger weight, it has the bolt. If both weight the same, the bolt is in either in the other group of 3 or the 1 bag left behind. 4--> CASE1: If the bolt is in one of the groups of 3 we weighed. We can find out the required bag by the same method. CASE2: If the bolt is either in the other group of 3 or the 1 bag left behind then we need 2 more steps to be sure:- 5--> Divide the 4 remaining bags into 2 groups of 2 bags. Weigh and find out the groups having the bolt. 6--> Weigh the remaining 2 bags and find out the thief .

This was the same method I used, however, I believe the thought process behind the question might be impeded by the provided answers.

This would've been a better question had it been a different style, such as "The minimum number of weighings = a and the maximum number of weighings = b. Find a + b".

This would've made it a much better mental problem, rather than one which can be solved by very simple trial and comparison of the given answers.

Alex Panebianco - 7 years, 3 months ago

ok :P. pardon me if u can. :D

D K - 7 years, 3 months ago

No worries, I just thought it might be a good suggestion, incase you wrote any more, similar problems. :)

Alex Panebianco - 7 years, 3 months ago

Shourya Pandey
Mar 17, 2014

There can't be any maximum or minimum in the problem. Athena is suggesting how many times must he have to weigh. So she will always take into consideration the worst possible case. Anyways, the answer should have been 6.

Zeus's Master bolt

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