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Puzzles and Conudrums 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. 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 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, ie., 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. Closed Door Probability Puzzle (also known as the Monty Hall problem)This is a fabulous demonstration of the power of faith in random decision-making over simple logic and probability. It was inspired 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 one-in-three-chance, 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, 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 'counter-intuitive', 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 1The door you originally chose was a one-in-three chance - ie, the likelihood of your guessing that door to be the winning door was one-in-three. The 'other' door is now a one-in-two chance, and the likelihood of your guessing the 'other' door to be the winning door is one-in-two. You are 50% more likely to correctly guess a one-in-two chance than a one-in-three chance, so pick the other door in preference to your original choice of door. Explanation 2The door you originally chose was a one-in-three chance. The other two doors collectively represented a two-in-three chance. When one of these doors is eliminated, the two-in-three odds transfer to the other remaining door, which you should now chose in preference to your original door, which was a one-in-three 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 one-in-twenty, to '50:50' evens, or (depending on how you look at it) arguably nineteen-in-twenty. 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 99-to-1 shot compared to the alternative which is at worst 50:50, and arguably a massive 99% chance? The Prisoner's DilemmaThis 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). 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 co-operation): There are two players or teams. Each has two cards, one marked 'Defect', the other 'Co-operate'. 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:
Try it. 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 co-operate. 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 tit-for-tat 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 short-lived. Pretty soon they'll get a reputation for being selfish and no-one will play them, let alone co-operate. 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:
Average Speed ConudrumA man travels 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? **** Answer: 40mph? Wrong. The correct answer is 30mph. Most people's instinct is to add the 20mph and the 60mph and divide by two (20 + 60 = 80. 80 ÷ 2 = 40), which is the way most averages are calculated. But this is not the way to calculate the average of two speeds over a given distance. The correct answer can only be found by adding the distances, then adding the times, and then calculating a new average: ie., 25 miles + 25 miles = total distance of 50 miles. 75 minutes and 25 minutes = total time of 100 minutes. 100 minutes = 1.6666 (recurring) hrs. 50 miles divided by 1.6666 hrs = 30 mph More resources? Click here for a library of new ideas! FAQs/Contact Us/Motivational & Inspirational Quotes/Useful Sites |