August 12, 2011

Missing dollar riddle

I was asked the following question last thursday:

You want to buy a T-shirt that costs RM97. You have no money.

You borrow RM50 each from your brother and your sister and with your current RM100, you purchase said T-shirt.
You hold in your hands the balance (RM3). You then decide to return RM1 to your brother, RM1 to your sister, and keep the remainder.

Now that each of them has RM1 back, the total owed is RM98. If you have RM1, what happened to the remaining RM1?

This riddle illustrates problems of confusion and misdirection can foil a person's clear understanding of the problem.

The original problem is as follows:

Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 for himself.
Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $27. The bellhop has $2. If the guests originally handed over $30, what happened to the remaining $1?

I shall attempt to solve the original question.

Explanation 1:

The initial payment of $30 is accounted for as the clerk takes $25, the bellhop takes $2, and the guests get a $3 refund. It adds up. After the refund has been applied, we only have to account for a payment of $27. Again, the clerk keeps $25 and the bellhop gets $2. This also adds up.
There is no reason to add the $2 and $27 – the $2 is contained within the $27 already. Thus the addition is meaningless. Instead the $2 should be subtracted from the $27 to get the revised bill of $25.
This becomes clearer when the initial and net payments are written as simple equations. The first equation shows what happened to the initial payment of $30:
$30 (initial payment) = $25 (to clerk) + $2 (to bellhop) + $3 (refund)
The second equation shows the net payment after the refund is applied (subtracted from both sides):
$27 (net payment) = $25 (to clerk) + $2 (to bellhop)
Both equations make sense, with equal totals on either side of the equal sign. The correct way to get the bellhop's $2 and the guests $27 on the same side of the equal sign ("The bellhop has $2, and the guests paid $27, how does that add up?") is to subtract, not add:

           $27 (final payment) - $2 (to bellhop) = $25 (to clerk)

Explanation 2(or the TL;DR version):

The question attempts to confuse you by adding up loaned money. Simply put, the $2 is contained within the $27, and should not be added together.

An analogous situation would be adding 99 empty pails and 1 full pail of water. The two quantities are distinct from each other and it is meaningless to call the total "100 full pails of water" nor "100 empty pails of water".

The wikipedia entry elaborates further:


The "paradox" cleverly sets its room rates so that when we add the two terms $27 and $2, we nearly get $30. If not for this "near-miss", we would be more inclined to ask if those two terms have to add up to $30 when we break down the situation this way (and to realize that they do not).
With different prices, the illusion would vanish. Say the clerk initially accepted $30 but then learned that rooms are only $10 no matter how many people are in them, and sends back a refund of $20 via the bellhop. Again, the bellhop, seeing that $20 doesn't evenly divide, gives each guest $6 (for a total of $18) and keeps the leftover $2 for himself. Therefore each of the three guests paid $4, bringing the total paid to $12; add that to the bellhop's 2 dollars to get a total of $14. So where did the other $16 go?
With this setup it is more clear that the guests' new total amount paid ($12) is only the bellhop's $2 away from the actual room price of $10, not the original room price of $30. The target price to account for is the new $10 bill, not the old $30 one. In the original riddle it is only the "near-miss" with $30 that makes $30 seem like the correct target of the operation.
The riddle involves the phenomenon of 'suspension of disbelief' inherent in storytelling and its power over the human imagination. If one were to make the story a bit more complex and compelling the illusion is almost guaranteed to work in the moment of its telling and can be a good illustration for the explanation of the anomaly, although not a perfect one because there is an explanation. The more points added to the story cause the listener to pause and try to compute what each element may signify.
There are dozens of variations to the riddle.

Have a nice day.

See also:


August 2, 2011

Humble Bundle 3.0 -Review- Part II

For our next title, we have...

4. Hammerfight

Hammerfight is about 2D battles of flying machines equipped with various slashing, piercing and blunt weaponry.A unique combat system is based on realistic physics simulation, and it ties the movements of the rider to the movements of your mouse. As you wave the mouse, your rider swings his warhammer, smashing the foe into the wall!

Basically, if you don't play this title, you're missing out on a lot. You feel the strikes as though they are real, and landing a powerful blow is immensely satisfying. It is amazing how the developers managed to simulate the mass of the weapon in your hands, and the physics involved is spot on.

 Hands down my favourite out of the bunch, but the others are still really good.

The controls are sharp and intuitive, and when paired with the realistic physics and smashable objects scattered across the levels and the high quality art, produces an explosion of fun of the highest caliber.

Did I mention that there's multiplayer too?

5. And Yet It Moves

And Yet It Moves is a 2D platformer with a twist. It brings the genre to another level by allowing the player to rotate the game world in 90 degree increments. So simple, and yet, so refreshing.

Your objective in each level is merely to reach the goal. There are no enemies to jump on, power-ups to use, or coins to collect. Conflict is created by the fact that your goal is not immediately accessible. A sheer cliff cannot be scaled -- until you spin the world 90 degrees clockwise, that is, creating a nice flat plain to stroll across.

A spin might give you access to new areas but also send loose rock tumbling down on your head to crush you. You may have to spin the world multiple times in order to manipulate a group of bats into scaring off a man-eating lizard. Playing And Yet It Moves, you're constantly discovering new things. There is a lot of trial and error, but regular checkpoints keep frustration levels down.

The challenges scale in difficulty as you progress. This title is a very pleasing and relaxing experience and is definitely worth a try.

And that's all the five titles in Humble Bundle 3.0. Also, you get free access to Minecraft for free until the fourteenth of August 2011. As Minecraft is not, strictly speaking, physics-related, you'll have to go somewhere else for the review.

Steel Storm is also given away for free if you buy the package. Awesome.

I've devoted 2 days of my student life for this review.

TL;DR: GET HUMBLE BUNDLE 3.0 if you're looking for games that are worth your time.