## 1998, Part 2

This is the continuation of the submitted puzzles supplied by Philippe Fondanaiche (Paris, France). Feel free to send partial solutions.
1. Many centuries ago, 6 thieves were arrested, among them the son of the King. Each of them was assigned a number (from 1 to 6). The King annouced to the ringleader that he intended to take clement measures: "Tomorrow morning, the 6 thieves will be transfered to the jailyard and placed in a circle. I will free my son who has the digit 2. Then I'll count clockwise a number of places equal to this digit and I'll free the man pointed out. I'll count again clockwise the number of positions mentioned by the new man's number. And so on... If I arrive at the place of a man already freed, all the remaining thieves will be put to death." The ringleader gave the problem a great deal of thought. The next morning, all the thieves escaped with their life. What was the layout imagined by the ringleader? What should be the layout to release all the thieves if their number is respectively 7,8,9,10,.......,1998?

2. Is there an integer N such that 1998*N = 22222.......22222 (only the digit 2 in the expression of this number)? If so, how many digits are in N?

3. A pocket calculator is broken. It is only possible to use the function keys: + , - , =, 1/x(inverse function). All number keys and the memory funtion work. How can we calculate the product 37 * 54? (The result is obviously 1998.)

4. What are the terms following the first terms of these sequences?
1. 11111001110, 2202000, 133032, 30443, 13130, .....
2. 238, 918, 1998, 3478,......
3. 24, 70, 118, 258, 494,......

5. What are the integer sides of the triangle such that the perimeter and the area are multiples of 1998, and the area is minimum?

6. Let's consider 2 circles of radius 1, the first one (C1) is tangent to the the x-axis at the origin and the coordinates of its center are (0,1); and the second one (C2) is tangent to C1 and to the x-axis, and the coordinates of its center are (2,1). We build C3 tangent to C1, C2 and the x-axis; then C4 tangent to C2, C3 and the x-axis; .....then Cn tangent to Cn-2,Cn-1 and the x-axis. What is the abscissa of the center of C1998?

Sources: Pierre Tougne (Pour la Science) (puzzles 1,2,3,8,13), Philippe Fondanaiche (puzzles 4,5,11,12), or derived of many collections of mathematical puzzles

Solution
Mail to Ken