Puzzles & Brain Teasers
Sudoku
The name Sudoku comes from syllables selected from a Japanese phrase, the whole phrase being loosely translated as: "The numbers must occur only once".
In the above solution (adding the red numbers) to the original puzzle (the black numbers), every row, evey column and every 3 by 3 square contains all of the digits 1 to 9, each occurring only once.
The squares need not contain numbers. It's the same game if you replace each digit with any other token, such as a kind of fruit or the picture of a well-known movie star.
For more information, see the following links.
Information About Sudoku
- Everything you could possibly want to know about Sudoku
- In other words, the very comprehensive article on Sudoku in Wikipedia.
It tells you more than you probably want to know about the game, including many strategies for solving it.
- Sudoku links
- This is a large, active set of links about Sudoku, from the Open Directory Project.
- Many strategies for solving Sudoku
- Here you will find a big list of solution strategies, including a few that are intended for automatic solving by computer, and are therefore completely pointless if you like solving puzzles yourself!
- The Sudoku Solver
- This is a major piece of work by Andrew Stuart.
- Don't get put off by the advanced stuff (even some of the so-called simple stuff is pretty advanced) - it's best to start solving Sudoku using simple approaches.
- However one of the fascinations of Sudoku lies in the discovery of its unexpected subtleties, which this site will really help with.
- Brian's Guide to Solving Difficult Sudoku
- A guide that I wrote to help myself, but other people may find it useful.
Play Sudoku
OK, so you know about the strategies for solving Sudoku, and you want to find some puzzles to solve!
Here are a few suggestions:
- Sudoku 2Go Pro App
- This is a great App with zillions of games, many variations, and an excellent progressive hint system which will soon let you progress from “easy” to “fiendish” puzzles.
- What I learnt using this App I have written up as Brian's Guide to Solving Difficult Sudoku, which you may find useful.
- Web Sudoku
- The most popular Sudoku site on the Internet, created by Gideon Greenspan and Rachel Lee.
- You can generate any number of Sudoku games, selecting your level of difficulty.
- Once you have a game on the screen, you can print it and play it offline, and/or you can complete it online (with electronic pencil marks allowing several tentative entries per cell, if you want) and check each move as you go along.
- Important:
- Select the Options button before the first time that you start to play!
- JigSawDoku
- A Flash-based presentation of essentially the same game, also produced by Rachel Lee and Gideon Greenspan.
- This is more fun than the first one in some ways (e.g. you can substitute symbols for numbers, and see and move all the unused numbers or symbols). However it doesn't have pencil marking or the same flexibility on the hint system.
- Other Web Browser Based Sudoku
- These links will pull up a lot more options to look at!
Free Online Puzzle Games
If you are looking for free online Sudoku puzzles, see above.
However, there are lots more! For example...
BLOXORZ
This game is simple in concept (like many of the best games) but is great fun and totally addictive.
A 2-by-1 block is moved across a tiled floor by rolling it sideways or lengthwise ninety degrees at a time.
The aim is to get the block to fall into the square hole at the end of each stage.
There are 33 stages to complete. You can return directly to any stage that you have already reached.
Bridges and switches are located in many levels. The switches are activated when they are first pressed down by the block.
Soft switches (round-shaped) are activated by any part of your block.
Hard switches (X-shaped) need the block to be standing on end in order to get enough pressure to activate them.
Some switches open or close bridges alternately. Some switches will only ever make certain bridges open, but won't close them.
Orange tiles are fragile. A block standing vertically on one may make the tile give way.
Teleporting switches, shaped like ( ), teleport your block to different locations, splitting it into two smaller blocks. These can be controlled individually and will rejoin into a normal block when both are placed next to each other.
Small blocks can't activate heavy switches, and can't pass through an exit hole without being rejoined into a complete block.
That's it! You can play the game here...
Lots More Free Online Puzzles
If you are looking for free online Sudoku puzzles, see above.
For a pictorial index to lots of other free online puzzles, go here.
It you prefer normal text descriptions, go here.
You will also find lots more free games of all kinds at FreeWorldGroup.com.
Have fun!
Brain Teasers
This section contains two great problems in logical thinking.
You don't need to be sitting in front of a keyboard in order to wrestle with these!
The first one is a pure logic problem, which totally killed all productivity in the place where I was working for a couple of days. We got there in the end... but it's a toughie.
The second one is a great example of how counter-intuitive the notion of probability can be.
I also introduce another problem in probability, which you can have some fun with if you haven't met it already.
The Lying Blackfoot, The Truthful Whitefoot... and the Random Greyfoot
If I remember correctly, this first appeared as a prize puzzle in the English Sunday Times, around 1970.
It is a diabolical variation on a simpler puzzle, which we need to look at first:
- The Simpler Version
- A jungle island is inhabited by two tribes, the Truthful Whitefeet (who always tell the truth) and the Lying Blackfeet (who, as you suspect, always lie). Unfortunately apart from their feet they look pretty much the same, and since they always wear moccasins, you can't tell them apart.
- Also, they get distinctly annoyed if you ask too many questions, which is not a good thing in these parts.
- Also, they only answer Yes or No to anything you ask them, and they will ignore you (and get annoyed) if you ask a question for which Yes or No wouldn't be a sensible answer, or indeed address any other kind of remark to them.
- An explorer arrives at a fork in the path, at which is standing such a tribesperson. The explorer is desperate to find the village which he knows is nearby. If he takes the wrong fork, he will probably perish from one of the numerous perils that lurk in this jungle, including dying of thirst.
- He knows that he had better ask just one question (to be on the safe side) in such a way that whether the answer is Yes or No, he will know which fork in the path will take him to the village.
- He could, of course, point to one of the two paths, and ask: "Is that the way to the village?" But he could have the bad luck to be talking to a Lying Blackfoot, and this is no time to be depending on luck.
- He knows that he had better not ask the tribesperson to take off his (or possibly her) moccasins either, in order to get a readout on the colour of the feet, since this will severely annoy him (or possibly her)... which is known not to be a good thing to do.
- After much cogitation, he comes up with a question. The tribesperson grumpily answers "No". "Thank you," says the explorer, and with much relief takes off down the correct path to the village.
- What was the question that he asked?
- If you know the answer, carry on! If you don't, try to work it out... and if you can't (please try first!) you will find the answer here.
- The Diabolically Hard Version
- A very similar jungle island is inhabited by three tribes, the Truthful Whitefeet (who always tell the truth), the Lying Blackfeet (who always lie) and the Random Greyfeet who simply answer Yes or No at complete random (they must toss a mental coin, or something... anyway they have no malicious or benign intent when they say Yes or No, it really is random, OK?). Unfortunately apart from their feet they look pretty much the same, and since they always wear moccasins, you can't tell them apart.
- These tribespeople have very similar characteristics to those on the other island, except
for two differences:
- They are just a little bit less easily annoyed than the tribespeople on the other island (but don't push your luck!).
- They always move around in a threesome, one person from each tribe (maybe because it isn't safe to let a Random Greyfoot wander around on his (or possibly her) own).
- Our intrepid explorer once again encounters a fork in a path, and he is in the same mess as he was before. Standing next to the fork is a threesome of tribespeople (the explorer knows that this must consist of a Blackfoot, a Whitefoot and a Greyfoot, since they always travel around together).
- His safest option is to ask just one question to one of the three tribespeople, but unfortunately the question he used before won't work, because he could have the bad luck to pick the Random Greyfoot, and whatever his question was, the answer wouldn't tell him reliably what he needed to know.
- Sweating profusely, he realises that he is going to have to push his luck a bit, and ask two questions. He could ask the same tribesperson two questions, or maybe put one question to one of the tribespeople and the second question to another of the tribespeople. Anything more, and they will all get severely annoyed... which would not be a good thing.
- He starts to panic... and then (because he has an extremely logical brain, however foolish he might be in choosing to wander around these jungle islands) the solution comes to him.
- He chooses one of the tribespeople (going "eeny... meeny... miny... mo..." under his breath) and asks his first question. "Yes," grumpily answers the tribesperson. He then asks his second question, and the tribesperson that he asks glowers threateningly and wonders if this scrawny object could possibly be good to eat, but finally answers "No". "Thank you," says the explorer, and with much relief takes off down the correct path to the village.
- What were the two questions that he asked?
- If you work out the answer, or you have tried hard to work it out but need some help, I would be only too pleased to hear from you - please e-mail me!
- Let me reassure you, though: there is an answer, and it is not based on gimmicks, just pure logic like the simpler version is. Also, whatever the explorer's solution is, it doesn't depend on luck (other than testing the temper of the tribespeople) - if he gets answers at all, then he is guaranteed to get the information he needs.
- Answer To The Simpler Version
- The key to this version of the puzzle is that the explorer has to get the same answer, irrespective of which tribe the person comes from. Finding the key of the puzzle is really important (as it will be with the harder version)!
- The explorer points to one of the two paths at the fork (the incorrect one, as it turns out), and asks: "If I were to ask you if that was the way to the village, would you say Yes?"
- A Truthful Whitefoot (let's assume he's male) knows he would say No if asked, and has to tell the truth about this, so he answers No, meaning that he would not say Yes if asked.
- A Lying Blackfoot (let's assume he's male) knows he would say Yes if asked the same question (he would lie), but has to lie about this fact, so he answers untruthfully No, meaning that he would say No if asked (which is a lie). Both tribespeople would be forced to answer the same question with the same answer, because the Lying Blackfoot is tricked into lying about how he would lie, and so ends up giving the same answer as the Truthful Whitefoot - which is what is wanted in this desperate situation!
- If that makes sense to you, you're ready for the harder version!
The Monty Hall Problem
This problem is associated with the American television game show artist, Monty Hall - hence its name. It is a great example of how poorly we understand the notion of probability.
The game
In the game there are 3 doors. Behind one of them is a car, behind each of the other two is a goat. If the contestant picks the door with the car, he wins it (we'll assume the contestant is male, just for convenience).
The contestant first picks a door (1, say). Monty (who knows where the car is) then opens another door (3, say), revealing a goat behind it. Monty then asks the contestant whether he now wants to change his initial choice.
Should the contestant change his original choice? There are two doors left, the car must be behind one of them. It appears to almost everyone that there is a 50-50 chance of either door being right, so most people assume that there is no virtue in changing.
Most people (including some eminent mathematicians in the past) are wrong. You are twice as likely to win if you change your original choice.
But why?
A simple explanation of the correct strategy
Here is the explanation in as simple a form as I can make it. But before you read on, do give it some thought yourself, if you haven't already.
OK...
We need to go back to the beginning of the game, when all the doors were shut. You pick a door (1, say). What is the probability of the car being behind it?
If you think "1 in 3" then you are correct. At this stage of the game, with all doors shut, you have 3 choices and only one of them is right. OK so far?
There are now two situations we need to look at after Monty opens the door:
- Your initial choice was correct. The car is behind it, and the other unopened door has a goat behind it.
- Your initial choice was wrong. A goat is behind it, and the other unopened door has the car behind it.
We don't know which situation we are in, but we must be in one of these two situations - right?
Remember that you are twice as likely to have made the wrong initial choice as the right initial choice - that's important.
So now Monty asks you, "Do you want to change your initial choice?"
In the first situation, you are bound to lose if you change your mind.
In the second situation, you are bound to win if you change your mind.
But (and here is the whole trick): you are twice as likely to be in the second situation as in the first situation.
That is because you made your initial choice when all the doors were shut, so your initial choice had a 1-in-3 probability of being right.
So you should change, because you are twice as likely to be in the situation where it is a good idea as you are to be in the situation where it is a bad idea.
Hold on, I still don't get this...
Even when we get to this stage, our brains are still trying to play tricks on us. What happened to our mental hangup that there is a 50-50 probability of either of the two remaining doors being right?
Let's imagine changing the game. In this different game, Monty starts by opening one of the doors, revealing a goat behind it. Now he asks you to pick one of the doors (1, say). What is the probability of the car being behind it?
(Yes, it's 50-50, or 1 in 2.)
Now Monty asks you (in this different game): do you want to change your initial choice?
You still have the same two possible situations as you did above. The difference is that now, in this different game, you are just as likely to be in the first situation (your initial choice was right) as in the second situation (your initial choice was wrong). So in this different game, it doesn't matter whether you change or not, you still have the same chance of winning.
If you compare the two different games and think about the differences between them, it might (or might not!) help. One thing is for sure: if you have problems with this puzzle, then you are not alone.
If you're still having problems, try this way of thinking about it:
When the contestant makes his initial choice, he has a 1 in 3 chance of being right.
When Monty opens one of the other doors, this does not change the contestant's chance of being right with his initial choice - OK?
But what has changed is that now the only other choice is the door that is still closed.
Since there is a one-third probability that the initial choice is correct, and there is only one alternative left, the probability of that other choice being correct must be two-thirds, or 2 out of 3.
This is because the probabilities of the only two choices left must add up to 1, i.e. to total certainty. It is absolutely certain that one of the two doors has the car behind it.
I think that the explanations I have provided above, even if they aren't as clear as they might be, are easier to understand than most other explanations you will find - but only you can be the judge of that!
You will find more descriptions of the problem, and the controversy that it still creates, here.
For a full treatment of the Monty Hall problem in Wikipedia, go here.
I still don't believe it...
If you still don't think that the "always change your mind" strategy works, or can't see why it works, then you're in good company! Some very learned (and not-so-learned) folks have spent a lot of time explaining why the strategy is wrong - and they all forgot to do one simple thing:
They forgot to actually do the experiment, to prove their theory one way or another.
So why not find a partner and play the game yourself? It's quite fun and instructive. You can easily simulate the game using a deck of cards. The Ace of Spades, for example, can represent the car, and the Jokers can represent the goats.
One of you plays "Monty", the other the "contestant". While the "contestant" shuts his eyes, "Monty" lays the cards out in a row in some order, face down, remembering where the Ace is. A card that is face down represents a closed door.
The "contestant" points to a card, but doesn't turn it over. This is like choosing the door.
Now "Monty" turns over one of the other two cards, picking one which he knows is a Joker (a goat). This is like opening a door and showing that there is a goat behind it.
The "contestant" now follows his preferred strategy, either always changing his mind or always NOT changing his mind, and turns over the selected card.
If you keep playing this game a few times, it won't be long before you find that the "always change your mind" strategy wins twice as often as the "don't change your mind" one.
A last thought...
Testing one's theories is part of "the scientific method". This experiment was a simple and fun example of something that is quite serious. It is a sad and scary fact that religious fundamentalism in the USA and elsewhere is trying to subvert and suppress the true basis of science, since it threatens belief systems that run counter to the knowledge that science has painstakingly and self-critically assembled since the time of the ancient Greeks.
But that's another story.
The Birthdays Scam (Part 1)
Our poor intuitive grasp of probability has led to many scams.
Here is the basis for a good one:
A party is getting started. People are arriving in the room. As each person arrives, he or she is assigned a "bogus birthday" which is randomly assigned from any of the possible 365 days in a normal year (not a leap year).
In this random assignment, there is nothing stopping the same birthdate being chosen more than once. Imagine the dates being written on 365 folded slips of paper in a jar, and each slip being returned to the jar after being chosen and all the slips being shuffled again.
(This is to avoid the problem that real birthdays might have a slight bias towards certain days of the year.)
How many people have to enter the room before it is more likely than not that two of them will have the same "bogus birthday"?
Our intuition might tell us that the answer is 182 people (about half of 365). If so, our inuition would be wildly wrong!
I can tell you that the answer is a number that is less than 30 - but I won't tell you yet how much less. Can you work out what the number actually is?
(The answer isn't much different if you use real birthdays instead of bogus randomly-assigned birthdays, by the way.)
If you can work out the answer, then you will see that it doesn't take many more people to arrive after that before the probability of two people having the same birthday goes way up... hence the opportunity for a betting scam - or a party game, come to that.
The Birthdays Scam (Part 2)
Did you try to get the answer yourself?
Here, if you need it, is a method of working it out.
It is based on the fact that the probabilities combine by multiplication, not addition, which is easiest explained by example:
In the Birthday Scam, after each person arrives the probability of them NOT having the same birthday as those who have already arrived is:
(The probability of the second person NOT having the same birthday as the first)
(The probability of the third person NOT having the same birthday as the previous two)
(The probability of the fourth person NOT having the same birthday as the previous three)...
...which becomes (364/365) x (363/365) x (362/365) x ... and so on.
If you start working this out on a calculator (which is pretty tedious) then the probabilities start at 0.99726... (very high probability of NOT having the same birthday) but gradually get smaller, the difference each time becoming greater.
If, like me, you use a spreadsheet to do the calculations, you will quickly discover that when the 23rd person arrives at the party, the probability of them NOT having the same birthday as the prevous arrivals is 0.4927..., or below 50%, which is the answer to the question originally posed. And by the time the 40th person arrives, there is only a 1 in 10 chance of no matching birthdays.
Mechanical Puzzles
If you like mechanical puzzles, then my job here is pretty easy. Just follow the first link below!
- Rob's Puzzle Page
- This is an absolutely superb site for all things to do with mechanical puzzles, lovingly produced by Rob Stegmann.
- When you're there, you might find his site map very useful - not only to guide you around, but because of its classification of various kinds of puzzles.
- Rubik's Cube
- This is a comprehensive article from Wikipedia devoted to what is possibly the most famous mechanical puzzle of them all.
- If you like this kind of thing...
- My "Did You Know?" page boldly goes where some remarkable discoverers have gone before, and tries to make sense of the amazing things that they found.
- Why do sunrises and sunsets not start to get earlier and later on the same date, after the winter solstice?
- What does Conway's Game of Life tell us about the real world?
- What is really going on in the awesomely beautiful and mysterious world of the Mandelbrot Set?