## 1998, Part 1

We've been in this year for quite awhile now, but this collection of puzzles supplied by Philippe Fondanaiche (Paris, France) is one of the best I've seen. Some of these are easy and some are quite involved. Feel free to send partial solutions. Part 2 will be posted next week.
1. With the four digits 1, 9, 9, 8 used in this order and considered separately or juxtaposed (i.e. 19 or 998) and the symbols + , - , x , / , sqrt, ! , ^ , (...), build up the longest possible list of integer numbers starting from 1, without interruption. As an example:
1 = (1^9)*(9-8)
2 = 19 - 9 - 8
3 = 1*sqrt(9) * (9-8)
etc..

2. With the 9 digits 1, 2, 3, 4, 5, 6, 7, 8, 9 used in this order and considered separately and the symbols above mentioned, we can express 1998 as follows:
1998 = (1+2) * 3/4 *(5*6 + 7)*8*sqrt(9).
Find with the minimum of symbols the 9 equations expressing 1998 by the successive cancellation of one digit from 1 to 9, the other 8 digits being used in the ascending order.

3. With the above mentioned symbols, express 1998 by using one digit, that means 9 different formulas to be built up. As an example, we can calculate 2048 with the digit 4 as follows:
2048 = sqrt(4)*4*4^4
Try to use the smallest number of symbols needed (i.e. not 1+1+...+1=1998).

4. Find a, b, c, d positive integers (d>c>b>a), all < 10000 such that:
1/a + 1/b + 1/c + 1/d = 1/1998 in the following cases:
1. (d-a) maximum
2. (d-a) minimum
3. a+b+c+d maximum
4. a+b+c+d minimum
5. a*b*c*d maximum
6. a*b*c*d minimum

5. Is it true that 11111^99999 + 99999^88888 is divisible by 1998? Same question with 111111^999999 + 999999^888888 ?

6. 1998 cards are numbered from 1 to 1998 and are placed in a circle in this order. Beginning with card 2, we eliminate the cards placed in an even position, continuing around the circle until there is one card. What is the number of this card?
Let's consider now a pack of N cards. With the same process above described, the remaining card is numbered 1998. What is (are) the possible value(s) of N? What if we choose every other card, starting with the first card?

7. 1998 cards are numbered from 1 to 1998 and are placed in a line in this order. The cards are face down (i.e. the numbers are invisible). In a first step, we turn over all the cards, that means all the numbers become visible. In a second step, we turn over 1 card out of 2 starting with the second card. In a third step, we turn over 1 card out of 3 starting with the third card, and so on...... In the 1998th step, we turn over the 1998th card. What are the visible numbers at the end this process?

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