## Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

# gavinksong

Member Since 09 Jul 2013
Offline Last Active Yesterday, 05:36 PM

### Mirror Primes

30 October 2014 - 11:24 PM

How many primes are the between 9 and 100 such that reversing the digits yields another prime?

This by itself is way too easy, so try to answer this using as little brute force as possible (or try to lower the upper bound as much as possible).

### Count the Flags

24 October 2014 - 11:05 AM

(Hello, friends. This is yet another puzzle from BWOC. I don't know the solution to this one yet, so I was thinking we could work on this together.)

You are tasked with designing a robot to explore a large but finite maze. The maze is drawn on a square grid, and walls exist on some of the edges of the square grid. Some of the squares contain a flag.

Your robot may interact with the world in the following ways:

1) Check which of the 4 adjacent edges contain walls.

2) Move to one of the 4 adjacent squares (provided there is no wall in the way).

3) Check if there is a flag on your square.

4) Pick up a flag (provided there is a flag on your square and the robot is not already holding a flag).

5) Put down a flag (provided the robot is holding a flag and there is not already a flag on your square).

6) Generate a random bit.

7) Output a number.

Your robot will be placed in a maze. The maze will contain some number of flags (from 100 to 1000). All flags will be reachable from the robotâ€™s starting position. Your robot is tasked with determining the number of flags. The robot may take as long as it needs, but may only output one number and must output the correct answer eventually, with probability 1.

The catch is that your robot is not Turing complete. It only has a finite amount of memory. You can give your robot as much memory as you need, but it must succeed on arbitrarily large mazes

### Digit Patterns

17 October 2014 - 03:49 PM

This is a variant of BMAD's recent post on the probability of choosing a natural number with a 1 in the digits.

From the set of all natural numbers...

1) What is the probability of choosing a natural number whose digits sum to an even number?

2) What is the probability of choosing a natural number whose digits sum to a number divisible by N?

### Risk Battles

16 October 2014 - 06:14 PM

In the popular board game, Risk, players try to occupy as much territory as possible by moving around their armies and attacking territory owned by other players. These battles are settled through a series of dice rolls.

The attacker rolls up to three dice, while the defender only rolls two. First, the highest values rolled by each player are compared. If the attacker rolled a higher value, the defender loses a unit. Otherwise, the attacker loses a unit. Then, the second highest values are compared in the same manner and one of the players loses a unit. This goes on until one of the armies becomes depleted.

If the attacker has fewer than three units, he may only roll the same number of dice as the number of units. Likewise, if the defender has fewer the two units, he may only roll one die.

1) What is the probability that a dice roll results in a draw?

Okay, so now pretend there is a battle where the attacker has n units, and the defender has m units.

2) What is the probability that the attacker wins the exchange?

3) What is the average/best/worst case running time of the entire exchange?

### Octodad Tries To Walk Straight

16 October 2014 - 05:45 PM

(This puzzle is based off of a student-created problem on an exam for an undergraduate Intro to Discrete Mathematics course at UC Berkeley)

Octodad has trouble walking straight. When he takes a step, he moves one yard in any random direction.

He decides to try and practice moving around on an xy-coordinate plane. If he starts at the origin, what will be the mean and standard deviation of his distance from the origin after taking a large number of steps?