The Frozen Lake

You wake up on a frozen lake in an isolated region, a hundred meters away from the shore. The surface of the lake is frictionless, and no grip of any kind can be attained over it. You find just your mobile phone in your pocket, but when you take it out to call for help, you realize there is no reception.

If there is no wind force to help you escape, what are you going to do to avoid freezing to death?

Throw your phone as hard as you can. Thanks to Newton’s third law of motion and the frictionless lake, you will start sliding away.

Mini Chess

White to play and force the black king to d3.

1. Qc3+ Ka2 2. Qc1 Kb3 3. Qa1 Kc2 4. Qa2+ Kc3 5. Qb1 Kd2 6. Qb2+ Kd1 7. Qa2 Kc1 8. Qb3 Kd2 9. Qb1 Kc3 10. Qa2 Kd3

If 4. … Kc1, then 5. Qb3 and we get to the position in move 8. If 4. … Kd1, then 5. Qb1 Kd2 6. Qb2+ and we get to the position in move 6.

On Fire

Slylock Fox and Max Mouse needed the rope on the ground to escape, but none of the onlookers could throw it as high as their window. However, Slylock and his sidekick found a solution and managed to escape the fire. What did they do?

Slylock asked the fisherman to cast his line to their window. After the fishing line was in Slylock’s hands, he told the beaver to remove the remaining line from the reel and tie it to the end of the rope. Slylock used the line to pull the rope up and then went down along it.

FEATURED

10 Dots, 10 Coins

If you have 10 dots on the ground, can you always cover them with 10 pennies without the coins overlapping?

Assume the dots lie in a plane and the radius of a penny is 1. Make an infinite grid of circles with radii 1, as shown on the picture, and place it randomly in the plane.

If we choose any point in the plane, the probability that it will end up inside some circle of the grid is equal to S(C)/S(H), where S(C) is the area of a coin and S(H) is the area of a regular hexagon circumscribed around it. A simple calculation shows that this ratio is larger than 90%. Therefore, the probability that some chosen point in the plane will not end up inside any circle is less than 10%. If we have 10 points, the probability that neither of them will end up inside a circle is less than 100%. Therefore, we can place the grid in the plane in such a way that every dot ends up in some circle. Now, just place the given coins where these circles are.

Source:

The Mathematical Intelligencer, 34:3 [September 2012], 11-14