99 unique numbers between 1 and 100 are listed one by one, with 5 seconds pause between every two consecutive numbers. If you are not allowed to take any notes, what is the best way to figure out which is the missing number?
SOLUTION
Keep up adding the given numbers and remember only the last two digits of the sum. In the end, if the result is less than 50, subtract it from 50. If the result is larger than 50, subtract it from 150. This method works because the sum of all numbers from 1 to 100 is 5050, so if you know the sum of all the listed numbers, you will know the missing number as well.
Below you can read the steps of a magic trick, as well as a video of its live performance. Your goal is to figure out how the trick is done, then perform it for your friends and challenge them to figure out the trick themselves.
Take out from your pocket a deck of cards, which is visibly shuffled.
Ask your first assistant to cut the deck, then take the top card from the bottom pile of cards and memorize it.
Ask your second assistant to take the next card from the bottom pile and memorize it.
Ask your first assistant to return his card back on the top of the bottom pile, then ask your second assistant to do the same.
Place the two piles of cards on top of each other and cut the deck multiple times.
Split the deck into two piles of cards, dealing consecutively one card on the left, then one card on the right, and so on, until you run out of cards.
Take one of the two piles of cards, look at it, and guess correctly what cards were chosen by your assistants.
How does the magic trick work? Below you can see a live performance of the magic trick from Penn and Teller’s show Fool Us.
SOLUTION
The secret of the trick is to memorize the group of cards which are located in even positions and the group of cards which are located in odd positions in the original deck. An easy way for doing this is to split the cards into two groups, such that the cards in the first group are only spades and diamonds, and the cards in the second group are only clubs and hearts.
When the two assistants pick their cards and then return them back into the deck, the order of the cards is reversed. When you split the original deck into two piles (even after cutting it several times), each of the piles will contain a card which should not be there. For example, the group of spades and diamonds will contain one clubs card, and the group of clubs and hearts will contain one diamonds card. These two cards are the ones which were picked by the assistants.
The diving bell that Slylock Fox and Max Mouse are using has snapped its chain. A rescue team can reach Slylock and Max within 20 minutes, but there are only 15 minutes of air left. What will Slylock do?
SOLUTION
He will use the lug wrench to remove the sunken car’s tires and release their air into the diving bell.
100 guests go to a party and some of them shake hands with each other. Show that there are two guests who handshake the same number of people.
SOLUTION
Each of the people at the party has shaken hands between 0 and 99 times. However, if someone has shaken hands 0 times (with nobody), it is impossible that another one has shaken hands 99 times (with everybody). Therefore there are at most 98 different options for the number of handshakes at the party, and thus two of the guests have shaken hands the same number of times.
On the image below you can see 11 points in the plane placed in such a way that there exist 6 lines passing through 4 points each. Can you place 10 points in the plane, such that there are 5 lines passing through 4 points each?
The first four moves played by White are 1. f3, 2. Kf2, 3. Kg3, 4. Kh4. If White gets mated in the fourth move, what could be the moves played by Black?
An eighty-year-old person was living alone in his house. On a sunny Friday morning, while delivering some letters in the mailbox, the postman felt suspicious and decided to look through the window of the house. That’s when he saw the old man lying dead on the floor in a puddle of blood. When the police arrived, they found three bottles of milk, the Tuesday’s newspaper, and some gifts in front of the house. The lock of the door was broken, and they already suspected a person. Whom and why?
SOLUTION
The police suspected the newspaper boy. Since there were 3 bottles of milk, the old man must have been dead for a few days already. However, the last newspaper left in front of his house is from Tuesday. Probably the newspaper boy, knowing that the old man is dead, decided that there is no point to go to his door and deliver newspapers anymore.
King Arthur and his eleven honorable knights must sit on a round-table. In how many ways can you arrange the group, if no honorable knight can sit between two older honorable knights?
SOLUTION
The answer is 1024 ways, up to rotation around the table. To see this, note that the youngest honorable knight must sit right next to King Arthur – there are two possible places for him. Then, the second-youngest knight must sit right next to this group of two. Once again, there are two possible places for him. Continuing like this, we see that for all honorable knights, except for the oldest one, there are two possible spots on the table. Multiplying two to the power of ten out, we get 1024.
