Ten Problems

Since Percy posted the 1=2 thing a few days ago, I’m tempted to make up something of my own. So here it is…

To proof:

proof:

multiply both sides by , we have

From property of logarithms, we have

Therefore, , and

What went wrong?

Update: Solutions are posted!

How does ??

Update: Solutions are posted!

This is really fun problem that I found today. The solution really eluded me for a long long time.

Hare and Tortoise had to take a make-up class in math over the summer, a two-month, self-paced course with a test at the end of each of 12 chapters. The course requires a 70% grade to pass.

In the first month, they both had difficulties with the concepts. Hare averaged 60% on his exams; Tortoise averaged 50%. Owl, the supervisor, spent three days of the next week helping them with their difficulties.

It worked. In the second month, Hare averaged 90% on his exams; Tortoise averaged 80%.

However, Tortoise got a passing grade of 75% in the class; Hare failed with 65%. When Hare protested to Owl that he’d outscored Tortoise in both months, Owl made Hare do the math on the board — and sign up for another make-up class, after school in the fall.

How did Tortoise pass while Hare flunked?

Source

Update: Solutions are posted!

This paradox is named after its originator, William A. Newcomb, a theoretical physicist at the University of California’s Lawrence Livermore Laboratory.

There are two boxes on the table: one opaque and one transparent. The transparent box has a dollar bill in it. The opaque box is empty at the moment. You have two choices: take the opaque box only or take both boxes.

One hour later, both boxes are removed. A computer called the decision prediction machine predicts the choice you have made. From experiments, the machine has 99% chance of predicting you decision correctly.

If the prediction machine predicts that you will take the opaque box only, a thousand dollars will be put into the opaque box. On the other hand, if it predicts that you will take both boxes, the opaque box will be left empty.

The boxes are returned to the table and you pick the box(es). Note that you have 99% chance of getting $1000 by picking the opaque box only. On the other hand, you always get $1 more by taking both boxes, regardless of the contents in the opaque box. What choice would you make?

Update: Solutions have been posted!

Three grandmothers, with their two granddaughters;
Two husbands, with their two wives;
Two fathers, with their two daughters;
Two mothers, with their two sons;
Two maidens, with their two mothers;
Two sisters, with their two brothers;
Yet only six in all lie buried here;
All born legitimate, from incest clear.
How might this happen?

Source: 536 Puzzles and Curious Problems by Dudeney

Update: Solutions have been posted!

When does Oct 31 = Dec 25 make sense?

Update: Solutions are posted!

Three men checked into a hotel. They each paid $10 for their rooms. Afterwards, the hotel manager tells the cashier that there was supposed to be a $5 discount, and gives her $5. The cashier gives each man $1 and keeps the other $2. So if each of the men paid $9, the cashier kept $2, and the men were supposed to have paid $30, what happened to other dollar?

Update: Solutions are posted