Dragos Velicanu
Some Cool Problems
1. Using only basic arithmetic operations, + - * / , and all the provided numbers exactly once, make the number 24:
a) 10 10 4 4
b) 7 7 3 3
c) 1 3 4 6
2. You are given a number of balls and a scale. Out of these balls all will be identical except for one.
a) You have 9 balls and one of them is heavier than the rest. Using the scale only twice how do you identify it?
b) You have 13 balls and one of them is either heavier or lighter, you don't know which. Using the scale only three times how do you identify it?
c) You have 12 balls and one of them is either heavier or lighter, you don't know which. Using the scale only three times how do you identify it and whether it's heavier or lighter?
3. Using only 4 straight lines and without lifting your pencil inbetween, connect all the dots:
o o o
o o o
o o o
4. Four people are on one side of a bridge at night. They have only one flashlight and a maximum of two people can be on the bridge at once. Whoever is crossing needs to have the flashlight. If one person can cross the bridge in 1 minute, another in 2, another in 5, and the last one in 10. How do you get everyone across in 17 minutes?
5. Ten people get captured by an evil wizard. He tells them that he will put them in a line one behind the other so that they can only see what is in front of them and will put a hat on each of their heads that is either red or blue, nobody can see their own hat. He will then proceed to ask them, in order starting from the person at the very end of the
line, what color hat he's wearing, and he is allowed to say either "red" or "blue" and may not convey any other signals. If the person guesses correctly he's free to go, otherwise he disappears and gets turned into a frog. The wizard lets them devise a strategy beforehand. What do they do to save the most number of people?
6. The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A
flea is sitting on top of the stake eyeing the kangaroo (hungrily). The kangaroo sees the flea
leaps into the air and lands one mile from the stake (with its tail still attached to the stake by
the rubber band). The flea does not give up the chase but leaps into the air and lands on the
stretched rubber band one inch from the stake. The giant kangaroo, seeing this, again leaps
into the air and lands another mile from the stake (i.e., a total of two miles from the stake).
The flea is undaunted and leaps into the air again, landing on the rubber band one inch further
along. Once again the giant kangaroo jumps another mile. The flea again leaps bravely into
the air and lands another inch along the rubber band. If this continues indefinitely, will the flea
ever catch the kangaroo? (Assume the earth is flat and continues indefinitely in all directions.)
7. You are walking around with a helium baloon attached on a string and you hop on a subway. After the doors close and the subway starts to accelerate forwards, does the baloon float towards the back of the subway, stay motionless, or drift forward relative to the subway car?
8. A cork submerged in a bucket of water will take a certain amount of time to float to the top. If everything is dropped from a plane and is in free fall, how does that time compare? Neglect air resistance.
9. You have many cut up pieces of string. You know that if you light one end on fire each piece will burn up in one hour, but they may not necessarily burn at a uniform rate.
a) How do you measure 30 minutes?
b) How do you measure 45 minutes?