puzzles and conundrums
complex puzzles, conundrums and exercises
Puzzles, conundrums and lateral thinking exercises help team building, motivation, and will warm up any gathering. Puzzles and conundrums like these are great brain exercises, and are good illustrations of how the mind plays tricks. See also the puzzles and tricks page, the team building games section for team building and employee motivation ideas, and try the expressions derivations quiz. Puzzles and lateral thinking exercises add interest to meetings and training sessions. Giving groups or teams a mixed set of puzzles is a great way to get people working together and using each other's strengths.
These lateral thinking exercises and complex puzzles are great for making people think, opening minds to new possibilities, and illustrating how the mind plays tricks and the importance of using the brain, instead of making assumptions.
boy girl probability puzzle
A man has two children. One of them is a boy. What's the probability that the other one is a boy? Most people think this is the same as: A man has a child, and it's a boy. He then has another child. What is the probability that the second one is a boy too? The answer to that is 1 in 2, or 50%. It's like saying: if I toss a coin and it comes down heads, what are the odds that it will come down heads next time? But this problem is quite different. This is why: If a man has two children, he could have had: first a boy, then a girl  BG These are the only possible ways in which a man can have had 2 children, and they are all equally probable. The probability of each is therefore 1 in 4, or 25%. Now let's come back to the problem. We know that at least one of the children is a boy. That cuts out just one of the 4 possibilities listed above  GG  leaving 3 possibilities: BG, GB, BB (each of which, as we've seen, is equally likely). Of those 3, only one  BB  satisfies the condition that the other child is a boy as well. So the answer is 1 in 3, or 33.3% (.3 recurring to be precise). Quite a lot of people refuse to believe this because they think that if a man has two children, he can have two boys, two girls or one of each, i.e., three possibilities, each equally likely. But they're not equally likely, because 'one of each' is not one possibility, but two. Think of all the people you know with 2 children, and you'll find that roughly a quarter of them have two boys, a quarter have two girls, and half of them have one of each. (Ack Terry Moran) 

closed door probability puzzle (also known as the Monty Hall problem)
This is a fabulous demonstration of the (commonly destructive) power of faith in random decisionmaking over simple logic and probability.
Aside from being an entertaining statistical curiosity, the puzzle demonstrates how readily many people base decisions their on instinct, and in many cases then reinforce their decision because of an entrenched personal stance in which logic and reason are utterly ignored  'denial' being a common term for the behaviour.
The puzzle was made famous by the format of an old USA TV gameshow 'Let's Make A Deal', hosted by Monty Hall, and the conundrum is commonly referenced in various forms by scientists and writers when demonstrating widely ranging aspects of probability theory and how the mind works.
Broadly the game show conundrum was:
A contestant is shown three closed doors. Behind one door is $10,000; behind the other two is nothing (or in most versions of the story, so presumably true, a goat). The contestant is invited to choose one of the doors and keep whatever lies behind. This offers a oneinthreechance, ie., 1/3 or 2:1, of winning. That is to say the contestant had a one in three chance of picking the right door.
When the contestant had chosen a door, Monty Hall then opened one of the other doors to reveal nothing behind it (or a goat), which left two closed doors, one with the money behind it and one with nothing (or a goat). The contestant was then asked if they wish to change their original choice (see note below), which creates the conundrum:
Have the odds changed as to which door wins and which door loses? And if so how?
What would you do? Keep your original choice, or choose the other door?
Most contestants on Monty Hall's show were apparently reluctant to change their original choice for fear that it was right, or because intuitively they felt that probability could not be altered by revealing one of the 'losing' doors. The problem is called 'counterintuitive', because the answer seems for many to defy instinct and logic, even after it's been explained several times.
You should change your choice, and here are a couple of ways of justifying why (mathematicians and probability experts can provide plenty more complex explanation than this if you need it):
explanation 1
The door you originally chose was a oneinthree chance  ie, the likelihood of your guessing that door to be the winning door was oneinthree. The 'other' door is now a oneintwo chance, and the likelihood of your guessing the 'other' door to be the winning door is oneintwo. You are 50% more likely to correctly guess a oneintwo chance than a oneinthree chance, so pick the other door in preference to your original choice of door.
explanation 2
The door you originally chose was a oneinthree chance. The other two doors collectively represented a twointhree chance. When one of these doors is eliminated, the twointhree odds transfer to the other remaining door, which you should now chose in preference to your original door, which was a oneinthree chance.
Logic and the law of probability says to switch original choice and pick a different door.
If you're still in doubt, imagine there are 20 doors  one has the money, the others nothing. You pick a door, then 18 doors are opened revealing nothing, leaving your choice and the one other door. Would you change your choice now? By switching doors you'd improve your chances from oneintwenty, to '50:50' evens, or (depending on how you look at it) arguably nineteenintwenty. Do you change your choice? It's staggering how many people still refuse to.
Still sceptical? How about 100 doors: Pick a door. Open 98 revealing nothing, leaving two doors, one a winner and the other a loser. Would you still prefer your original 99to1 shot compared to the alternative which is at worst 50:50, and arguably a massive 99% chance?
More technical explanation at http://www.io.com/~kmellis/monty.html
If you choose to ignore the first stage then of course it's an equal 1 in 2 chance, but the point of the puzzle is that there was a first stage when the odds were against you. You can decide to ignore that information, but ignoring it won't change the original odds. The odds of your initially picking correctly a oneinthree or oneinahundred chance do not change just because you reveal the losing options. This is the central point of the puzzle. What about starting with a million doors? If you life depended on it? Would you still say it's a 50:50 chance and risk sticking, knowing that if you stick with your original choice you would be backing a oneinamillion shot? Try it with a pack of playing cards and and friend. Not with a million cards of course  a more manageable number  and you will see that sticking with the first choice is successful on average broadly in line with the starting odds.
Note: While the following points of interest do not alter the main principle of the puzzle in any way, I have received various correspondence as to the nature of choice offered to contestants after opening the first door in the Monty Hall Show. It has also been suggested to me that Monty Hall said in an interview that he did not recall offering contestants the chance to change their minds. If you know more about this please tell me.
This interesting insight is from R Krob (Feb 2005)  "Monty Hall did not offer the contestants a chance to change their minds about the door (only). What he did offer them was the chance to have another chance at something else, for example: Contestant chooses the third door. The first door is opened to reveal a goat. Monty would then offer he contestant a choice between their original door or what was (hidden) in a box. The box could contain another goat or a prize of less value than the main prize (for example furniture), and sometimes the 'something else' would be a known 'sure thing', for example: Contestant chooses the third door. Door one is opened to reveal a goat. The contestant would then be offered a choice between their original door or what was in the box, which would be revealed before their decision. Maybe grand prize was a truck and box prize was kitchen appliances." (Ack R Krob)
This contribution is from Yvan Auffret (October 2006) concerning a TV show in Brazil several years ago which seems to have borrowed and developed the Monty Hall puzzle format. It was described to him by a Brazilian professor of statistics Ademir José Petenate, who apparently now teaches in Campinas, Brazil and uses the show example in his classes. Yvan writes "... there is indeed a game show that is exactly what you are describing. It's one Ferrari and two goats, and if you lose you get to spend one week with the goat (followed by camera crew)..."
(Thanks also to Rupert Stubbs, Wendy Ramsay and John for suggestions and guidance.)
the prisoner's dilemma
This amazing model was based on a traditional gambling game, and brought to prominence particularly by the American political scientist Robert Axelrod. The prisoner's dilemma has been studied and discussed for decades by strategists, gamblers, philosophers, and evolutionary theorists (see note about Richard Dawkins below). It represents a fundamental aspect of thoughtful choice over unquestioning instinct, a dilemma which we see played out everywhere  from the simplest human relationship to the largest global conflict.
At its core is the question of whether to collaborate or not (called 'defect' in this model).
The question is often is a measure respectively of emotional maturity and security (enabling collaboration and trust) versus personal insecurity and fear (tending to prompt aggression, animosity, conflict and defection).
Here's how the gambling game works. At a simple level it's great for demonstrating the dangers of selfish behaviour, and the benefits of cooperation:
There are two players or teams. Each has two cards, one marked 'Defect', the other 'Cooperate'. There is a neutral banker, who pays out or collects payments depending on the two cards played. Each player or team decides on a single card to play and gives it to the banker. The banker then reveals both cards.
Here's the scoring system:
 Both play the 'Cooperate' card  Banker pays each £300.
 Both play the 'Defect' card  Banker collects £10
 One of each card  Banker pays 'Defect' £500, but collects £100 from 'Cooperate'.
Try it. See the 'winwin' teambuilding game based on the prisoner's dilemma on the teambuilding games page.
The tendency if for each team to play Defect all the time, in hope of the big payout, and as a defence against being 'suckered' and having to pay the big fine. But where do these collective tactics lead? In the end the banker will collect all the money, albeit at £10 per round, but the banker always wins and both players always lose.
After a while, the players realise that their only hope for survival and beating the banker is to cooperate. Of course along the way, one or other players might be tempted to play 'Defect' and will collect the big payout having exploited the trust of the other side, but is this a sustainable strategy? Of course not. It reignites the titfortat aggressive defence scenario when both sides play 'Defect' and both sides lose.
Try playing the game with a group of people who randomly pair up for each round (single show of cards). Again, some players will attempt a strategy of continuous 'Defect'. Their gains however will be shortlived. Pretty soon they'll get a reputation for being selfish and noone will play them, let alone cooperate.
Professor Richard Dawkins, Fellow of New College, Oxford, provides more fascinating explanation about this model and just how fundamental it is to our existence in his book The Selfish Gene.
The model is called the prisoner's dilemma after the traditional story of two prisoners who are suspected of a crime and captured. The evidence is only sufficient to achieve a short custodial conviction of the pair, so they are separated for questioning, and each invited to betray his partner in exchange for their freedom. Not permitted to meet and discuss their decision, they each face the following prisoner's dilemma:
 Both men refuse to betray each other  Each receives a 6 month sentence due to lack of evidence.
 Both men betray each other  Each receives a reduced 1 year sentence because they told the truth.
 One betrays while the other does not  The betrayed gets 3 years while the betrayer goes free.
average speed conundrum
A man travel by car to a destination 25 miles away. The journey was made between 7 and 9am, so the roads were congested and progress was slow. The journey took 75 minutes, which means that his average speed was 20 miles per hour. He took the same route on the return and travelling in the middle of the day made faster time: his return journey lasted just 25 minutes, meaning that his average speed fro the return journey was 60 miles per hour.
What was his average speed for the two journeys combined?
euler puzzles
We've all seen these geometric puzzles which challenge you to trace along all of the lines, without lifting the pen from the paper and tracing each line only once. How many of these puzzles can you complete? How many of these puzzles are possible and how many are impossible? Is there a way of knowing whether a puzzle of this sort is possible without trying it? There is (see Euler's Rule). Try them first and then check the answers, as well as learn the secret.
When you know the Euler's Rule it's easy to make up your own puzzles like these, and be confident of whether they are possible or not  all you need is a pen and a flipchart: great for coffee break diversions, team building exercises, etc.
Puzzle 1  Easy ones first?......  
Puzzle 2  Still easy? Can it be done?....  
Puzzle 3  Looks easy, but not so easy. Can it be done?  
Puzzle 4  Now this one is easy...  
Puzzle 5  Easy.  
Puzzle 6  Easy.  
Puzzle 7  Looks very difficult. Can it be done?  
Puzzle 8  Don't even try this one...  
Puzzle 9  It's got something to do with where you start...  
Puzzle 10  Same applies here  where you start makes all the difference  assuming it can be done...  
Puzzle 11  Like I said, assuming it can be done...  
Puzzle 12  It would be a pity to end with one that's impossible... 
I call them euler puzzles after the man who devised the method of knowing if they are possible or not. Leonhard Euler was a brilliant 18th century mathematician who solved the puzzle of the Konigsberg bridges (now Kaliningrad, in Russia). Seven bridges connected both sides of the city and two islands in between. The townsfolk passed their time for many years trying (unsuccessfully) to find a route which would cross each bridge only once. Euler proved that is was not possible, until an eighth bridge was constructed (unfortunately little remains of the original city or its bridges since the 2nd world war). Nevertheless Euler's Rule survives, and is now used in planning routes and service territories (for postal services, meter reading, doortodoor deliveries, etc) the world over, to eliminate unnecessary travel and backtracking. So what's the secret?
For all the answers and Euler's Rule go here.
See also the amazing Mobius Strip puzzle.
More complex puzzles and conundrums will added from time to time, so please keep visiting this page. If you have a puzzle or conundrum of your own that you'd like featured on this page (with suitable acknowledgement of course) please send it.
More quicker puzzles and tricks are on the main puzzles page.
See also the great ditloids puzzles, the cliches and words origins quiz, and the team building games section.
The use of this material is free provided copyright (see below) is acknowledged and reference or link is made to the www.businessballs.com website. This material may not be sold, or published in any form. Disclaimer: Reliance on information, material, advice, or other linked or recommended resources, received from Alan Chapman, shall be at your sole risk, and Alan Chapman assumes no responsibility for any errors, omissions, or damages arising. Users of this website are encouraged to confirm information received with other sources, and to seek local qualified advice if embarking on any actions that could carry personal or organisational liabilities. Managing people and relationships are sensitive activities; the free material and advice available via this website do not provide all necessary safeguards and checks. Please retain this notice on all copies.
© alan chapman 19952014
Below is a solution to the Euler puzzle 12, if you need it, which should help you solve the others too. Try to work it out for yourself first using the rules above (i.e., start at one odd node, and finish at the other) before resorting to the full answer. Obviously start at point 1, which is the odd node, bottomright, then go to 2, then 3, and so on, finishing at point 27, the other odd node, topleft. 
Euler's Konigsberg Bridges puzzle analysis
The river banks and islands are nodes. The bridges are effectively lines connecting these nodes. Prior to the construction of the eighth bridge (shown on the left in dotted lines, the puzzle was impossible because there were four odd nodes (count the number of bridges from each of the two river banks  3  and two islands  5 and 3). When the eighth bridge was built the river bank nodes became even nodes with four bridge connections each, leaving only two odd nodes, and a solvable puzzle.  
Here's a network diagram of the same thing  a
topological map; the topology method of simplifying network analysis was also
invented by Euler. The nodes at A and B have been changed from odd to even with
the dotted line, which is the eighth bridge. (Incidentally, the famous London Underground Map is another example of a topological map  it's not geographically accurate at all, but it is very much easier to understand than a pure geographical map.) If you want to read more about this and other fascinating mathematical puzzles and explanations read 'Why Do Buses Come In Threes?' and the followup 'How Long Is A Piece Of String?' by Rob Eastaway and Jeremy Wyndham. 
See the Mobius Strip or Mobius Band puzzle
search businessballs website
browse categories


The use of this material is free provided copyright (see below) is acknowledged and reference or link is made to the www.businessballs.com website. This material may not be sold, or published in any form. Disclaimer: Reliance on information, material, advice, or other linked or recommended resources, received from Alan Chapman, shall be at your sole risk, and Alan Chapman assumes no responsibility for any errors, omissions, or damages arising. Users of this website are encouraged to confirm information received with other sources, and to seek local qualified advice if embarking on any actions that could carry personal or organisational liabilities. Managing people and relationships are sensitive activities; the free material and advice available via this website do not provide all necessary safeguards and checks. Please retain this notice on all copies.
© alan chapman 19952014