Puzzle : The Fake Coin Challenge!

There are 101 coins out of which 1 is fake, the fake coin is identical to a genuine coin but differs in weight. Using weight balance only twice how can we determine whether the fake coin is heavier or lighter than a genuine coin?

 

 

This is how to do it! 

Fake Vs Genuine Coin Weigh Comparison

This was the challenge!

Remember we are asked to determine whether the fake coin is lighter or heavier when compared with the genuine coin and not to identify the fake coin itself.

Keep aside any one coin. Divide remaining 100 coins into 2 groups of 50 coins each. Put these 2 groups on 2 pans of the balance.

1. If they weigh equal the the coin that is kept aside is fake. Weigh it against any genuine among 100 coins to know whether fake coin is lighter or heavier than genuine.

2. If they are not equal then that means the fake coin either made one side heavier or the other side lighter.

3. Take the heavier group of 50 coins for the next test. Divide them into 2 groups of 25 coins each. 

4. Put 25 - 25 coins on weighing balance. If they weigh equal then that means no fake coin among them which also means the fake coin was in the other group of 50 coins which was lighter in the first weighing. 

Hence, the fake coin is lighter present in the other group of 50 coins making the group slightly lighter compared to group of 50 genuine coins.

4.1. And if the result of weighing 25 - 25 coins is unequal then it's clear that the fake coin is among these 50 coins. Also, it must be heavier making this group to weigh more than the other group of 50 genuine coins in the first weighing.

This way, we can determine whether the fake coin is heavier or lighter than genuine one using the weighing balance only twice.

Find The Heaviest Ball

Given 27 table tennis balls, one is heavier than the others.

What is the minimum number of weighings (using a two-pan balance scale) needed to guarantee identifying the heavy one? The other 26 balls weigh the same.

Here is how to identify the heavier ball.

Identifying The Heavier Ball

 What was the task given?

We need to use the scale only 3 times.

1. Divide the 27 balls into 3 groups of 9 balls each. 

2. Use the scale to weigh 2 groups. This will tell us which group has the heavier ball. 

    If the scale is balanced then third group must be having heavier ball

3. Now, divide the 9 balls into 3 groups of 3 balls each and weigh 2 groups, and identify the group which has the heavier ball. Again, if the scale is balanced then the third group must be having the heavier ball.

4. From the heavier group, weigh 2 balls to determine the heavier ball. And, if the scale is balanced then the third ball must be heavier.

A Brillian Deception

A witch owns a field containing many gold mines. She hires one man at a time to mine this gold for her. She promises 10% of what a man mines in a day, and he gives her the rest. Because she is blind, she has three magic bags who can talk. They report how much gold they held each day, and this is how she finds out if men are cheating her. 

Upon getting the job, each man agrees that if he isn't honest, then he will be turned into stone. So around the witch's mines, many statues lay! 

Now comes an honest man named Garry. He accepts the job gladly. 

The witch, who didn't trust him said, "If I wrongly accuse you of cheating me, then I'll be turned into stone."  

That night, Garry, having honestly done his first day's job, overheard the bags talking to the witch. He then formulated a plan... 

The next night, he submitted his gold, and kept 1.6 pounds of gold. Later, the witch talked with her bags.

The first bag said it held 16 pounds that day. The second one said it held 5 pounds. The third one said it held 2 pounds. 

Beaming, the witch confronted Garry. "You scoundrel, you think you could fool me. Now you shall turn into stone!" the witch cried. One second later, the witch was hard as a rock, and very grey-looking.

How did Garry brilliantly deceive the witch? 

Here is the Garry's Master Plan!
 

Mathematics Behind The Brilliant Deception!

What was the deception?

As per the honest man, he must have mined 16 Pounds of gold since he kept 1.6 Pounds (10% of 16) gold for himself.

And as per magical bags, since Bag no.1 said it had 16 Pounds, Bag no.2 said 5 Pounds and Bag no.3 said 2 Pounds the total gold mined 16 + 5 + 2 = 23 Pounds.

We know, honest man Garry would never do any fraud and neither of magical bag would lie. 

So, it's clear that some of pounds are counted multiple times by magical bags. There are 23 - 16 = 7 Pounds gold extra found by those bags.

Now, Bag No.3 must have 2 Pounds in real.

And to make count of Bag No.2 as 5, Garry must had put 3 Pound + Bag No.3 itself in Bag No.2. Hence, the Bag No.2 informed witch that it had total 5 Pound of gold.

Finally, to force Bag No.1 to tell it's count as 16, Garry must had put 
11 Pounds + Bag No.2 (5 Pounds = 3 + Bag No.1) = 11 + 5 = 16 Pounds.

In short, Garry put 2 pounds of gold in Bag No.3 and put that Bag No.3 in Bag No.2 where he had already added 3 Pounds of gold. After that, he put this Bag No.2 in Bag No.3 where he had already added 11 Pounds of gold (somewhat like below picture).

 

This way, 2 Pounds of gold in Bag No.3 are counted 2 extra times and 3 Pounds gold of Bag No.2 are counted 1 more extra time. That is 2 + 2 + 3 = 7 Pounds of extra gold found by bags.

PUZZLE : The Case of Honest Suspects

Handel has been killed and Beethoven is on the case. 

He has interviewed the four suspects and their statements are shown below. Each suspect has said two sentences. One sentence of each suspect is a lie and one sentence is the truth. 

Help Beethoven figure out who the killer is.

Joplin: I did not kill Handel. Either Grieg is the killer or none of us is.

Grieg: I did not kill Handel. Gershwin is the killer.

Strauss: I did not kill Handel. Grieg is lying when he says Gershwin is the killer.

Gershwin: I did not kill Handel. If Joplin did not kill him, then Grieg did.

Click here to know who is the killer! 

DETECTION : Killer in The Case of Honest Suspects

What was the case?

Let's first see what are the statements made by 4 suspects.

Joplin: I did not kill Handel. Either Grieg is the killer or none of us is.

Grieg: I did not kill Handel. Gershwin is the killer.

Strauss: I did not kill Handel. Grieg is lying when he says Gershwin is the killer.

Gershwin: I did not kill Handel. If Joplin did not kill him, then Grieg did. 

ANALYSIS :

1] If Joplin is the killer then his first statement would be lie & other statement must be true. But his second statement contradicts assumption that he is killer. So Joplin can't be the killer.

2] Let's assume Grieg is the killer. His first statement must be lie and second must be true. His second statement points to Gershwin as a killer. But there is one killer among 4 so Grieg and Gershwin can't be killers together. Therefore, Grieg can't be the killer.

3] Suppose Gershwin is the killer. Again, his first statement is lie and second is true. His true second statement suggests that Grieg is the killer as Joplin is assumed to be innocent in the case. Again, both Grieg and Gershwin can't be killers together since there is only one killer. Hence, Gershwin is not the killer.

4] Therefore, Strauss must be the killer. His first statement is lie and second statement is true. And that's how Grieg is lying when he is saying Gershwin is the killer.

CONCLUSION : 

STRAUSS is the one who committed the crime and his first statement is lie and second is true.

The first statements of rest of all suspects are true and second statements of each are lie.