(Alice - 10) "Here is the baking pan," said the King, "so now you can make me the tarts." "Can't," said the Queen. "The recipe is in my cookbook, and the cookbook has just been stolen!". Well, the most likely suspect was the Duchess's Cook, and the cookbook was indeed found in the Duchess's kitchen. The only possible suspects were the Cook, the Duchess, and the Cheshire Cat.
"The Cheshire Cat stole it!" said the Duchess at the trial.
"On, yes, I stole it!" said the Cheshire Cat with a grin.
"I didn't steal it!" said the Cook.
As it turned out, the thief had lied and at least one of the others had told the truth. Who stole the cookbook?
Answer: Duchess
(Alice - 11) Shortly after the cookbook was returned to the Queen, it was stolen a second time. At the trial, they made exactly the same statements as the last time. Only this time, the thief lied and the other two either both lied or both told the truth. Who stole the cookbook this time?
Answer: Cook
Lottery Pairs
Starting in 2005, a group of M.I.T. students executed a daring lottery scheme, eventually inning over $3.5 million in a game called Massachusetts Cash WinFall. Central to their plan was an ingenious mechanism of avoiding risk by guaranteeing themselves a large payoff on every drawing. How did they do it? The basic idea can already be grasped in the following simplified example: Suppose you were playing a lottery where each drawing picked 3 out of 7 numbered balls. When a ball is picked, it isn’t replaced, so a drawing always consists of three DIFFERENT numbers, like 2,3,7 or 3,4,5. The order of the numbers doesn’t matter.
If you get all three numbers right, you win the jackpot, $6. If your ticket has two correct numbers out of three, you win a $2 consolation prize. One or fewer correct numbers and you get nothing.
Tickets cost 80 cents each. I want to buy a set of 7 tickets which guarantees me a profit on every drawing. (Since my tickets cost $5.60, this means that either one of my tickets needs to be a jackpot or I need to win consolation prizes with at least three of my tickets.) How should I pick my tickets?
Instructor Notes: Younger kids may play by having only 5 numbered balls, and they can buy 4 tickets.
Discussion: We know that of the three numbers in the draw, each pair (and there are 3 pairs in them) must appear on one of our tickets. So lets start from that. The total number of pairs are 7C2, i.e. 21. Since we have 7 cards and each card has 3 pairs (3C2), we also have 21 pairs. The trick is to make sure our 7 cards have every possible pair. Lets start to construct that. One rule we will observe is that we will not repeat any pairs (since we have 21 pairs to cover and only 21 options). Without any loss of generality, let the first card we pick be 1,2,3. Now we need other pairs with 1 so we can form two more tickets 1,4,5 and 1,6,7 - we have covered all 1-pairs. Now we need to complete pairs for 2 with 4,5,6,7 - Since we dont want pairs to repeat, we dont want 2,4,5 - so lets put 4 with 6 and have 2,4,6 and subsequently 2,5,7. Now we need 3-pairs with 4,5,6,7. Again to make sure pairs dont repeat, we can have 3,4,7 and 3,5,6. We are done! Check the answer to make sure every pair is covered.
For the smaller problem, 5 numbered balls means we need to cover 10 pairs. 4 tickets means we have 12 pairs available. So we can easily construct this, actually with some repetition. One example could be 1,2,3 1,4,5 2,3,4 and 2,3,5
Odd Problem
A pot contains 75 white beans and 150 black ones. Next to the pot is a large pile of black beans. A somewhat demented cook removes the beans from the pot, one at a time, according to the following strange rule: He removes two beans from the pot at random. If at least one of the beans is black, he places it on the bean-pile and drops the other bean, no matter what color, back in the pot. If both beans are white, on the other hand, he discards both of them and removes one black bean from the pile and drops it in the pot. At each turn of this procedure, the pot has one less bean in it. Eventually, just one bean is left in the pot. What color is it?
Answer: White. The cook only ever removes the white beans two at a time, and there are an odd number of them. When the cook gets to the last white bean, and picks it up along with a black bean, the white one always goes back into the pot.
Homework
In a far away land, it was known that if you drank poison, the only way to save yourself is to drink a stronger poison, which neutralizes the weaker poison. The king that ruled the land wanted to make sure that he possessed the strongest poison in the kingdom, in order to ensure his survival, in any situation. So the king called the kingdom's pharmacist and the kingdom's sage, he gave each a week to make the strongest poison. Then, each would drink the other one's poison, then his own, and the one that will survive, will be the one that had the stronger poison. The pharmacist went straight to work, but the sage knew he had no chance, for the pharmacist was much more experienced in this field, so instead, he made up a plan to survive and make sure the pharmacist dies. On the last day the pharmacist suddenly realized that the sage would know he had no chance, so he must have a plan. After a little thought, the pharmacist realized what the sage's plan must be, and he concocted a counter plan, to make sure he survives and the sage dies. When the time came, the king summoned both of them. They drank the poisons as planned, and the sage died, the pharmacist survived, and the king didn't get what he wanted.
What exactly happened there?
Answer: The sage's plan: He made a weak poison and drank it just before the test. During the test time, he would drink the stronger poison which he thinks the pharmacist would have made. Meanwhile, he takes a non-poisonous drink to the test and presents it as his poison. In this case the pharmacist would drink it and then drink his own poison and die.
The pharmacist's plan: He knows that the sage is going to drink a weak poison before the test and that he is going to present a non-poisonous drink. So the pharmacist counters this by presenting a non-poisonous drink too effectively disabling the sage from countering the poison he consumed earlier.