Fractal maze 3: Walls and carpets

See fractal maze 1 and fractal maze 2.  Following those other two examples, I wanted to try designing a maze that would be qualitatively different.  In all previous fractal mazes, lines represent paths.  Why can't lines represent walls, like in a traditional maze?  Because then you can't have paths cross over each other.  So I tried creating a fractal maze which does not have any crossing paths:

Click for a larger version

This maze contains four copies of itself, which for notation purposes I've labeled A, B, C, and D.  If you actually try to zoom in, you'll find that the lines become too thick.  Just pretend that the lines are mathematically thin.  In this maze, you must travel from the start to end without crossing any lines.  (No fair leaving the maze to do it!) 

If you'd like to check solutions, please e-mail me at skepticsplay at gmail dot com.

Fractal Maze 2: Sierpinski paths

Years ago I created a fractal maze, which was rediscovered by reader Nathaniel Arnest.  Nathaniel and I had a great e-mail discussion about fractal mazes.  Later I may post my thoughts on this subject, but for now I present a maze that we collaborated on.  Nathaniel designed the maze, and I chose the graphical design to be based on the Sierpinski Triangle.  This one is a bit easier than the other one (which is a good thing).

 Click to zoom in.

Rules:
1. Follow the colored paths from the empty light green circle to the filled dark green circle.
2. You may not travel along the black lines.  The black lines simply outline an infinite number of squares.  All squares are copies of each other, even when the lines are too small to see in the picture (except for the start and finish, which only occur in the largest square).
3. You may not jump from one color to another, except when passing through the boundary between squares.
4. The path you follow cannot go infinitely deep.  Finite solutions only!

If you'd like to check solutions, please e-mail me at skepticsplay at gmail dot com.  Describe the solution in any way you wish, or attach an image.

If any readers are having difficulties due to colorblindness, please e-mail me to suggest better colors to accommodate you.

Puzzle: Don't step on the grass

I was suddenly reminded of a geometric puzzle.  This puzzle is not original, but the thematic content is mine.

There is a patch of grass on a high school grounds.  It's shaped like a square (each side being one unit long).  The students aren't supposed to walk on it, but they do.  You could prevent them from walking on the grass by surrounding it with a fence.

But you can save on some fencing by exploiting a key fact about these high school students: they will only ever walk through the grass in straight lines.  So as long as you build enough fences to block all straight lines through the grass, you can prevent students from walking through it.  (Note that paths skimming the edge of the grass don't count, but paths cutting through the corners do count.)  What's the minimum length of fencing necessary?

If that puzzle is too easy, imagine that the grass is shaped like a regular hexagon.

I don't know if this will help, but there's a website which lets you construct geometrical shapes, as if you had a ruler and compass. Here is a square to start, and here is a hexagon.

I would have asked about a grass field shaped like a regular pentagon, but it's quite difficult.  Try it if you're brave.

See the solution

Draining a tank

This is a puzzle of my own creation.

There are nine identical 10-gallon tanks in a 3 by 3 square (the image shows the view from above).  One of the corner tanks is full of water, while the rest are empty.

You may open or close any of the walls between adjacent tanks.  If any walls are open, then the amount of water in all the connected tanks will equalize.  Assume that they equalize fast enough that you can't close walls in the middle of the process.  You may have multiple walls open at once.  For example, if you open all the walls, then each tank will have 10/9 gallons in it.

Your task, should you choose to accept it, is to open and close walls in such an order that you drain as much water out of the filled tank as you can.  What's the best you can do?

See the solution

Inner triangle area

I haven't posted a puzzle for a long time.  So here's one.

I start with any old triangle.  I construct the three red lines shown above by the following:
1. Take one side of the triangle and divide it into thirds.
2. Draw a line from the one-third point to the opposite corner.
3. Repeat for the other sides of the triangle.  Make sure to pick the appropriate one-third point so that it resembles the above figure (which is to scale).

Find the ratio of the area of the pink inner triangle to the area of the large black triangle.  Prove it.

See the solution

Fair Dice

There are three six-sided dice, with the numbers 1 to 18 on their faces.  If you roll any two of the dice against each other, each one is equally likely to "win" by rolling the higher number.  If you roll all three dice, they are equally likely to win.  No ties are possible.

One die has the numbers:

1, 5, 10, 11, 13, 17

Can you figure out the numbers on the other two dice?

This puzzle was inspired by a much harder puzzle submitted by Eric Harshbarger to MathPuzzle in April 2012, where there were four twelve-sided dice.

Folding into thirds

It's easy enough to fold a piece of paper in half.  Just fold one end onto the other.  But whenever I fold paper into thirds, I just guess where one third is and fold it there.

But there is a way to fold it into thirds exactly.  Can you figure it out?

(This is a relatively easy puzzle inspired by my recent attempts at origami. In the mean time, I've been trying to figure out if there is a way to fold an exact regular pentagon.  It seems that the standard way is to approximate.)

See the solution

No fault lines

Build a rectangle with multiple 2x1 bricks such that the rectangle has no "fault lines".  A fault line is a straight line through the rectangle that does not cut through any of the bricks.

For example, the following rectangle fails because of the fault line indicated in red.

What's the smallest rectangle you can make without fault lines?  You may send solutions to skepticsplay at gmail dot com.

Can you do the same with 3x1 bricks?

This puzzle is taken from Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon W. Golomb.

See solutions

The Car Parking game

Let's play a game.

(image borrowed from here)

I call this the car parking game.  There is a curb that can fit up to five cars on it.  We take turns parking new cars on the curb.  You may park your car anywhere on the curb where there is room.  That is, its position can be described by any number between 1 and 5 (inclusive), including fractions.  Cars take up some space, so the minimum distance between any two cars is 1.  So for example, it is not possible to park cars at 1 and 1.5.

If it is your turn and you have no space left to park a car, you have an excuse to double-park, and thus you win.  (I know this win-condition sounds backwards, but the game is more interesting this way).

So here's how we'll play.  Leave a comment, state which game you are playing, and state where you are parking your car.  I'll respond to your comment with my move.  You may try as many times as you want, so persistence can be a substitute for cleverness.

Game 1: The curb can fit up to 5 cars.

Game 2: The curb can fit up to 8 cars

Game 3: The curb can fit up to 17 cars (this one's for the programmers)

I posted some comments on strategy.

The relatively prime graph

Wow, it's been some time since I've posted a puzzle!  Here's a simple pure math puzzle off the top of my head.

Back in middle/high school, I would kill time in classes drawing graph of all the points (n,m) such that n and m are relatively prime.  Relatively prime means that there is no integer greater than 1 which divides both n and m.  The graphs would look something like this:

The black squares represent (n,m) where n and m are relatively prime, while the white squares represent (n,m) where n and m are not relatively prime.

The question is, can you find a 3x3 white square somewhere in this graph?  In other words, find N and M such that (N,M) are not relatively prime, nor are the eight surrounding pairs, (N-1,M-1), (N,M-1), (N+1,M-1), (N-1,M), etc.

It's not a particularly elegant problem, but think of it as open-ended.  There are many solutions, and many methods will work to find them.  Can you find one?

solution posted

Fillomino #2

A few months ago, I submitted a fillomino (aka polyominous) puzzle to a puzzle design competition on A Cleverly Titled Logic Puzzle Blog.  I think this is the second one I've ever designed (here's the first).

Rules of fillomino:
1. Divide the grid into polyominoes, which are connected shapes made from the little squares.  For example, tetris pieces are polyominoes made of four squares.
2. Fill each square with a number, representing the number of little squares in that polyomino.
3. No two polyominoes with the same numbers may share an edge.
4. Some of the numbers are given, but some polyominoes may be implied and have no given numbers.

Solvers voted on the puzzles they liked best.  Mine got second, which might reflect on the kind of people who voted.  Seriously, they clearly had a taste for rather difficult puzzles.

You may email solutions to skepticsplay at gmail dot com.  Enjoy your Thanksgiving!

The ant and the rubber band

In a draft I'm writing, I'm including a classic puzzle as a demonstrative example.  But I should let my readers have a try first!

There is an ant on a rubber band.  The ant is crawling from one end of the rubber band towards the other at 1 inch per second.  The rubber band is one foot long, but getting longer.  After each second, the rubber band's length grows by one foot.

Question: Will the ant ever reach the other side of the rubber band?

(It's an infinitely stretchy rubber band, and the ant lives infinitely long.  Since I am a physicist, I will not worry about whether the rubber band is stretching instantaneously every second or stretching continuously.  Just assume whichever you prefer.)

You have less than a week to solve this one.  Bonus points if you can predict what my draft is about.

(solution contained here)

Tower of Hanoi variant

The Tower of Hanoi is a classic puzzle that looks like this:

Image borrowed from Wikipedia

The idea is to get all the disks from the left rod onto the right rod.  This might seem easy (just dump them out and put them back on the other one), but there are a few rules you have to follow.  First, you can only move one disk at a time, and only from one rod onto another.  Second, no larger disk is allowed to be on top of a smaller disk.

The solution to the Towers of Hanoi is not too difficult, though the number of moves required increases exponentially as the number of disks.

Let's play a variant of the Towers of Hanoi.  Instead of three rods, there are five.  And there's an additional rule: a disk can only lay on top of another disk only if the one below is exactly one size bigger.

My question is: What's the tallest tower that you can move from one rod to another?

Afterwards, you can try the same variation with six rods.

(This is an original puzzle, inspired by too many games of Freecell, which obeys the same rules.  However, I would be surprised if I'm the only one who has ever thought of this variant.)

See the solution

Coins on a table

This is a game for two players.  We take turns placing quarters on a round table.  The round table has five times the diameter as the quarters.  When a player cannot place their quarter fully flat on the table (not on top of any other quarters) without disturbing any of the other quarters, that player loses.

Which player has the winning strategy?  What is the winning strategy?

(This is not an original puzzle.)

See solution

Ten rows of three

Here is a way to arrange eight dots into four rows of three.

Can you find a way to arrange nine dots into ten rows of three?

For the purposes of this puzzle, a row is just a set of distinct colinear points.  No two rows may lie on the same line.

See the solution

The rigged card game

I place nine cards on a table, 1 through 9, face down.  They are grouped in triplets.  First you pick a triplet, and then I pick another triplet.  Then we each reveal a random card from the triplet we picked, and the higher card wins.

I know what cards are in which triplets, but I don't know which random cards we will reveal within those triplets.  How can I arrange the cards such that, no matter which triplet you pick, I am more likely to win than you?

(This game is taken from an article by Martin Gardner called "Nontransitive Paradoxes", which attributed it to Leo Moser and J. W. Moon.)

See solution

The confused passenger

Classic puzzle:

An airplane has 100 passengers. But the very first one to get on is missing his ticket! He just takes a random seat, and refuses to move from that spot.

The rest of the passengers board the plane one by one. Each passenger takes the correct seat, unless it's already taken. If their seat is already taken, they sit in a random seat, and refuse to move.

What's the probability that the last passenger gets the correct seat?

(Hint: First try to consider an airplane with 2 or 3 passengers)

Extra Challenge:

It turns out that the first passenger isn't isn't really confused, he's malicious!  He knows where he's supposed to sit, and intentionally sits anywhere but there.  What's the probability that the last passenger gets the correct seat?

see the solution

The tiger and the lake

You are in the middle of a circular lake. At the edge is a tiger.

(Image credit)

In this puzzle, the tiger can't swim (which is funny, because when I was looking for a tiger photo, I found pictures of tigers doing just that).  Instead, the tiger will run along the edge of the lake, doing its best to catch you.  The tiger runs four times as fast as you can swim.
You want to get out of the lake, but if the tiger is waiting for you when you reach the shore, then the tiger will seriously damage your ability to not die.  Can you get out of the lake safely?  How?
This puzzle is a classic, not original.

See the solution

Logicians with hats

Three logicians play a game with colored hats. (That's always how the trouble begins.)

Each logician wears a hat, and that hat has a 50% chance of being red and a 50% chance of being blue.  The probabilities for each hat are independent of each other.  None of the logicians is allowed to see the color of her own hat.  They may make a game plan before starting, but during the game they may only look at the other hats without communicating.

And yet, the logicians guess the colors of their own hats.

They guess simultaneously, writing their guesses on paper which are only revealed after all guesses have been made.  They may write "red", "blue", or "abstain".  If at least one logician guesses right and none guess wrong, then they win.  Abstaining counts as neither right nor wrong.

What strategy should the logicians use to have the highest probability of victory?

This puzzle was taken from Haidong.  He also asks what happens if there are 2^k - 1 logicians.  The general case is rather difficult, but you should at least try for 7 logicians.

Solution posted

Heaven and Hell

Oooh, with a title like that, some people will be confused about this post's purpose.  This is a puzzle.

There once was a man who saw heaven and hell, but found they were nearly the same.  Each one consisted of a group of people sitting at a round table with food.  But their chopsticks were way too long to eat with!  The difference was that people in heaven were always full and happy, while the people in hell were always hungry and unhappy.  Why?

This puzzle comes from Stories to Solve, which is the other book I found from my childhood.  However, the book takes this story from an older tradition, so I don't feel I am stealing it.  Some people might categorize this as a lateral thinking puzzle.  I generally don't like lateral thinking puzzles, but I like this one.