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Mathematical Reasoning Puzzles

These puzzles require both logical and mathematical reasoning. Can you solve them?


A solo dice game is played where, on each turn, a normal pair of dice is rolled. The score is calculated by taking the product, rather than the sum, of the two numbers shown on the dice.

On a particular game, the score for the second roll is five more than the score for the first; the score for the third roll is six less than that of the second; the score for the fourth roll is eleven more than that of the third; and the score for the fifth roll is eight less than that of the fourth. What was the score for each of these five throws?



A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?



You must cut a birthday cake into exactly eight pieces, but you're only allowed to make three straight cuts, and you can't move pieces of the cake as you cut. How can you do it?



Can you place six X's on a Tic Tac Toe board without making three-in-a-row in any direction?

Tic Tac Toe Board



Nine dots are arranged in a three by three square. Connect each of the nine dots using only four straight lines and without lifting your pen from the paper.

Nine Dots



You want to hire a temporary employee for one month. You offer him reasonable wages, but the employee suggests an alternative. For the first day of work, he will be paid a penny. For the second day, two pennies. For the third day, four pennies. The salary for each subsequent day will be double the previous day's, until the one month term is over. Ignoring the legalities of such a situation, would it be a good idea to accept the potential employee's proposal?



You drive to the store at 20 mph and return by the same route at 30 mph. Discounting the time spent at the store, what was your average speed?



If you drive to the store at 20 mph, how fast must you go (again returning by the same route) for your average speed to be 40 mph?



Arrange the numbers 1 through 9 on a tic tac toe board such that the numbers in each row, column, and diagonal add up to 15.



Two trains travel toward each other on the same track, beginning 100 miles apart. One train travels at 40 miles per hour; the other travels at 60 miles an hour. A bird starts flight at the same location as the faster train, flying at a speed of 90 miles per hour. When it reaches the slower train, it turns around, flying the other direction at the same speed. When it reaches the faster train again, it turns around -- and so on. When the trains collide, how far will the bird have flown?