## Cut and Arrange

Cut the shape into two pieces, and then arrange the pieces into a square.

**SOLUTION**

The solution is shown below.

Cut the shape into two pieces, and then arrange the pieces into a square.

The solution is shown below.

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.

After you die, you somehow appear in a mystical room with two doors and two keepers inside. One of the doors leads to Heaven and the other door leads to Hell. One of the keepers is always lying and the other keeper is always saying the truth. If you can ask one of the keepers whatever question you want (you don’t know which keeper is lying and which one is truthful), how can you find your way to Heaven?

You can point your finger to one of the two rooms and ask any of the keepers the question “If I ask the other keeper whether this room leads to Heaven, would he say YES?”. If the answer is NO, go through that door, if the answer is YES, go through the other one.

Look carefully at the picture below and answer the questions.

1. How many tourists are staying at this camp?

2. When did they arrive: today or a few days ago?

3. How did they get here?

4. How far away is the closest village?

5. Where does the wind blow: from the north or from the south?

6. What time of day is it?

7. Where did Alex go?

8. Who was on duty yesterday? (Give their name)

9. What day is it today?

1. There are 4 people.

2. They arrived a few days ago, enough so that a spider web can appear on the tent.

3. Judging by the paddles, they got there with boats.

4. There is a hen walking around the camp, so the closest village is not far away.

5. The leaves of the trees are larger at the south side, so the wind must be blowing from the South.

6. The shadow is pointing towards West, so it must be morning.

7. Alex went to catch butterflies.

8. Since Peter is on duty today – cooking food for the group, it was Colin on duty yesterday.

9. Today is August 8. Watermelons ripen in August.

You have 9 balls, 8 of which have the same weight. The remaining one is defective and heavier than the rest. You can use a balance scale to compare weights in order to find which is the defective ball. How many measurements do you need so that you will be surely able to do it? What if you have 2000 balls?

First, we put 3 balls on the left side and 3 balls on the right side of the balance scale. If the scale tips to one side, then the defective ball is there. If not, the defective ball is among the remaining 3 balls. Once left with 3 balls only, we put one on each side of the scale. If the scale tips to one side, the defective ball is there. If not, the defective ball is the last remaining one. Clearly we can not find the defective ball with just one measurement, so the answer is 2.

If you had 2000 balls, then you would need 7 measurements. In general, if you have N balls, you would need to make at least log₃(N) tests to find the defective ball. The strategy is the same: keep splitting the group of remaining balls into 3 (as) equal (as possible) subgroups, discarding 2 of these subgroups after a measurement. To see that you need no less than log₃(N) tries, notice that initially there are N possibilities for the defective ball and every measurement can yield 3 outcomes. If every time you get the worst outcome, you will make at least log₃(N) tries.