Mental Exercise

What am I? When I am new, I hide my face. When I am full I show my face. At quarters I show half my face. What am I?

Tiler's Clue. Think about a floor that has been tiled with square tiles, each tile the same size. Usually, the installer of the tile starts in one corner of the room and proceeds out from there, laying new tiles next to the ones already laid. If you enter a room that has been tiled in this way, you can usually tell which corner the installer started laying tile. How?

Matching Birthdays. A professor was lecturing his class one day remarking on the fact that in a gathering of 23 or more people, the odds are at least 50% that two or more of those people share a birthday on the same day of the year. One observant student said that he would be willing to bet the professor even money that two of the students in the present class share a birthday. The professor remarked that in their small class of just 11 students the odds would be highly in his, the professor's, favor. The student said that he would still be willing to take the bet. Suspicious, the professor scanned the room and realized why the student would be guaranteed to win the bet. What could the professor and student seen to reach this conclusion. 

Good or Cheap. A restaurant posted a sign outside which presented two statements: Good food is not cheap. Cheap food is not good. Now, the question is, logically speaking, are these two statements equivalent?

What Profit. Suppose a person buys an item for $7 and then sells the item for $8. Later, the same item is bought for $9 and sold for $10. What is the person's total profit?

Medical Test. There is a medical test for a disease that has a 5% false positive (for every 100 people that test positive for the disease, 5 of them will not really have the disease). The incidence of the disease in the population is one in every one thousand people. Suppose you get tested for the disease and the test is positive. What are the odds that you have the disease? Hint: it is not 95%.

Pet Store. A pet store sells two kinds of birds, large birds and small birds. The large birds cost twice as much as the small birds. A lady comes into the store one day and buys five large birds and three small birds. Having a change of mind she changes her purchase to three large birds and five small birds. The bill for the original purchase was $20 more than the bill for the changed purchase. How much are the large birds and how much are the small birds?

The Dilemma. I can't say that there is an objectively right answer to the following puzzle. But I can say that there is a clever one. Why don't you give it a try? Suppose that while you are driving your car you happen upon a bus station one stormy night. You catch a glimpse of an old friend so you stop. You find that because of the storm, that three people, including your old friend, have been stranded: an old lady who is injured and in need of medical attention, an attractive woman who you obviously share a mutual attraction with, and the aforementioned old friend who would love to catch up with you. You would like to pursue the mutual attraction with the lady, but the old lady needs to be taken to the hospital right away. What do you do?

Answers

What am I? The moon.

Tiler's Clue. Usually, a room is not sized to perfectly fit an exact number of tiles in the horizontal or vertical direction. So three of the room's corners will have tiles that have been trimmed on one or more sides. But the corner where the tile installer started will have a full sized tile.

Matching Birthdays. The professor discovered that there were identical twins both present in the class (there is a small chance that identical twins could be born on separate days, one just before midnight, one just after midnight, but it would still be a good bet for the student).

Good or Cheap. Yes, they are equivalent. Both of them convey the information that no food exists which is both good and cheap.

What Profit. The total profit is $2. Perhaps the simplest way to see this is to consider that the purchases total $7 + $9 = $16. The sales prices total $8 + $10 = $18. The difference between these is $2.

Medical Test. As difficult as it may be to believe, the odds are only about 2% that you actually have the disease. Consider that a positive test could be the result of you actually having the disease or not having it but testing falsely positive for it. Let's find the odds for each of these: Real positive: .001 x .95 = .00095 False Positive: .999 x .05 = .04995 The real positive is much less frequent than the false positive, if you divide them it is only about 2% (0.0190.).

Pet Store. The original purchase differs from the changed purchase by the exchange of two large birds for two small birds. The $20 price difference is accounted for by the replacement of these two large birds for two small birds. This means that $10 is the price difference between a large and small bird. Given that the large birds cost twice as small, large birds are $20, small birds $10.

The Dilemma. With a little "out-of-the-box" thinking, you hit upon the idea of letting your old friend (whom you presumably trust) take the old lady to the hospital in your car while you volunteer to stay behind to "look out for" the woman.