[Guest]

Every now and then you see a puzzle, peek at the answer, then wish you'd taken the time to solve it instead of peeking. I did that with this puzzle, but I won't post the solution now so you don't make the same mistake. Enjoy!

A river has six islands in the middle, connected by 13 bridges as shown:

Enemy tanks are approaching on the other side of the river, so you've set up explosives to blow up the bridges. Unfortunately your detonators came from a dodgy batch, so each bridge has only a 50% chance of blowing up (independently of the others). What is the probability that the enemy tanks will still be able to find a route across the river?

If you find yourself doing any complicated calculations, you're barking up the wrong tree! Try a little lateral thinking instead...

EDIT: When you've solved this, there's a related, more general question to consider here...

[Guest]

well...the tanks have 50% chances for each and every bridge...so..50%?

[Guest]

Visiting this forum after nearly one year.

I enjoyed this one!

Please consider the possible outcomes.

For Simplicity, consider two islands, two shores, connected by five bridges (so that a H is formed)

There are 2^5 outcomes, each being an independent event. Either the bridge is blown or NOT blown.

The only way to cross is when:

1 is NOT blown AND 4 is NOT blown

1 is NOT blown AND 3 is NOT blown AND 5 is NOT blown

2 is NOT blown AND 5 is NOT blown.

2 is NOT blown AND 3 is NOT blown AND 4 is NOT blown

The truth table below shows if the bridges are blown= T means detonator worked and bridge is blown, whereas F indicates bridge is still intact.

(1) (2) (3) (4) (5) Result?

T T T T T Cannot Cross

T T T T F Cannot Cross

T T T F T Cannot Cross

T T T F F Cannot Cross

T T F T T Cannot Cross

T T F T F Cannot Cross

T T F F T Cannot Cross

T T F F F Cannot Cross

T F T T T Cannot Cross

T F T T F Can Cross

T F T F T Cannot Cross

T F T F F Can Cross

T F F T T Cannot Cross

T F F T F Can Cross

T F F F T Can Cross

T F F F F Can Cross

F T T T T Cannot Cross

F T T T F Cannot Cross

F T T F T Can Cross

F T T F F Can Cross

F T F T T Cannot Cross

F T F T F Can Cross

F T F F T Can Cross

F T F F F Can Cross

F F T T T Cannot Cross

F F T T F Can Cross

F F T F T Can Cross

F F T F F Can Cross

F F F T T Cannot Cross

F F F T F Can Cross

F F F F T Can Cross

F F F F F Can Cross

So, sixteen cross and sixteen cannot cross. That is 50%.

BTW, tanks crossing the bridge and boats sailing across are mutually exclusive only if we assume that the bridge is destroyed cleanly and does not impede the boat after it is destroyed. What Happens if the bridge is destroyed, but falls in such a manner that the boat cannot pass through? :-p just kidding ;-)

[Molly Mae]

Also, we must assume that all explosives detonate (or dud) at the same time before any tanks have begun crossing any bridges. But yes, the combined probability that a tank and boat will cross is 1, but only one of those events may happen.

[Guest]

There are definitely scenarios wherein both, the ship and the tanks, can cross. I sat and traced out a couple of them. Try it!

I'm not sure that I am correct but I got a 79.6875% chance of the enemy being able to cross over.

In case I take the simpler, 2 islands 5 bridges, scenario this reduces to 75%.

But I am certain that the logic for it being 50% is flawed because the inherent assumption is invalid

BTW: This is my 1st post ever so sorry if im doing something wrong.

Cheers!!

Edited December 22, 2010 by Harshj_89

[Guest]

There are definitely scenarios wherein both, the ship and the tanks, can cross. I sat and traced out a couple of them. Try it!

I'm not sure that I am correct but I got a 79.6875% chance of the enemy being able to cross over.

In case I take the simpler, 2 islands 5 bridges, scenario this reduces to 75%.

But I am certain that the logic for it being 50% is flawed because the inherent assumption is invalid

BTW: This is my 1st post ever so sorry if im doing something wrong.

Cheers!!

I'm sure that no scenario exists where both ships and tanks can cross. If that were the case the path of the ship would have to cross the path of the tank. Since the ships can't go over land or under intact bridges, and tanks can't cross water, their paths could not cross. I am intrigued as to what the examples you drew look like!

[dark_magician_92]

is 5/8 the answer ?????

[Guest]

There are 21 possible routes for the tank to take to traverse the river.

And there are 21 ways to block the path of the tank from crossing.

There is a 50% chance that one of those 21 routes will be passable.

And a 50% chance that one of the 21 ways to block the tank will occur.

So, the tank has a 50% chance of making it across the river.