Math Riddles

Answer: He will reach the end of the rope eventually. Once he gets onto the rope, the rope behind him also stretches, so this must also be accounted for.

Answer: The back prisoner will yell "black" if there is an odd number of black hats in front of him and "white" if there is an even number of black hats in front of him. The next prisoner will then count the number of black hats in front of him and if it was odd and is now even, or vice versa, then that person knows what color their hat is. The next person then knows their hat color based on what the people before them said and how many black hats are in front of them. In this way the front 99 prisoners will know their hat color and will be set free. The prisoner in the back who goes first has a 50 percent chance of being set free.

Answer: He can propose a plan that he gets 98 pieces of gold, the 3rd pirate gets 1 piece, and the 5th pirate gets 1 as well.

If there were just 2 pirates the younger pirate would definitely deny the plan so he could get all of the gold.

If there were 3 pirates the first pirate can offer the second pirate 1 piece of gold and take the rest himself because the second pirate wouldn't get anything if he has to propose a plan himself.

If there were 4 pirates the first pirate could take 99 for himself and offer 1 to the youngest pirate. They would both agree. If the youngest disagrees then he won't get any gold in the next plan.

So when there are 5 pirates it is in the interest of the 3rd and 5th pirate to accept 1 piece, because if they don't they won't get anything in the next plan.

Answer: There are two reasons:

First, Thomas is double counting a lot of the days. A lot of the time spent sleeping, eating, and relaxing occurs during weekends and the summer. Weekends also occur during the summer, so all of these hours are getting counted several times.

Secondly, school isn't an all day thing. So the 4 days actually represents more days of school. If school is 6 hours per day, those four days represents 16 days of school.

Answer: 43.

After 6 all numbers divisible by 3 can be ordered (because they can all be expressed as a sum of 6's and 9's). After 26, all numbers divisible by three when subtracted by 20 can be obtained. After 46, all numbers divisible by three when subtracted by 40 can be obtained. After 46, all numbers fit into one of these 3 categories, so all numbers can be obtained. 43 is the last number that doesn't fall into one of these categories (44 = 20 + 6 * 4, 45 = 6 * 6 + 9).