Help solve the Unsolved or Partially solved problems. These problems are from Ken's Archive - If you solve the problem and wish to submit the solution, please e-mail webmaster@g4g4.com or Ken(please visit Ken's Archive) Thank you for your interest.

The Length of Memory
In the game of Memory, there are many different cards, each having one matching card to make its pair. To start the game, all the cards are spread out face down. On each turn, a player chooses a card and turns it over, then chooses a second card and turns it over. If the cards match, they are removed from the play area. If the cards do not match, they are returned to their face-down positions. Assuming you have perfect memory (you can remember where all previously seen cards are), what is the expected number of turns needed to clear the play area for N pairs? How does this result change if two cards must first be selected, then turned over simultaneously? How does the first result change if there are mulitple identical pairs, such as in a standard deck of cards (any Queen could be paired with any other Queen)? In all cases, assume you play to minimize the number of turns. Source: Original.

Coloring a Cube
Coloring each face a solid color and using each of N colors (N=2,3,4,5,6) at least once, in how many ways can you color a cube, such that a duplicate cannot be found simply by rotating the cube? Using each of N colors at least once (N=2,3,4), in how many ways can you color a tetrahedron? The "Using each of N colors at least once" in the problem statements helps to limit the number of solutions. Can you solve them if that phrase is changed to "Using N colors"? Source: Several sources.

Trigons
.--A--.--B--.
/ \ / \ / \
C D E F G H
.--I--.--J--.--K--.
/ \ / \ / \ / \
L M N O P Q R S
.--S--.--T--.--U--.--V--.
W X Y Z a b c d
\ / \ / \ / \ /
.--e--.--f--.--g--.
h i j k l m
\ / \ / \ /
.--n--.--o--.

Trigons are similar to Triominos, except instead of having numbers at each corner, the numbers are on the sides of each triangular piece. Can you take a set of 24 trigons, consisting of all possible configurations of the values 0, 1, 2, and 3, and place them into a hexagon, two units on a side, such that each adjacent side matches correctly? Or, show why it can't be done.

(Note that 1-2-3 is a different trigon than 1-3-2, since neither can be rotated to create the other; while 1-1-2 would be the same as 1-2-1, since the latter can be rotated to obtain the former.)
Since I think the above problem may be easily solved, consider the following extension puzzles:
Try to solve the puzzle, making the sum at the 7 internal points the same (the sum of the six numbers surrounding the 7 internal points).
Try to solve the puzzle, making the six internal trigons have different sums. (The sum of the three numbers on each trigon.)
Source: Original, based on a puzzle in Dell's Math Puzzles and Logic Problems magazine.

Two Detectives
A crime was committed in Puzzleton, and 8 possible suspects were identified. Working separately, two detectives (Albert and Brutus) each narrowed down the list to two names. They know that between them, they have three names on their lists and the name that is shared is the actual culprit. The two detectives meet at the police station to compare notes and find out who the final suspect is, but they are not allowed to communicate unless a police officer is with them. How can they each learn who the culprit is without the police officer knowing? (The officer may know the culprit is one of two people, but should not have definite knowledge at the end of their conversation.) Can a method be found if there were originally more or less than 8 names? [How do they know they share a single name on their two lists? Why don't they meet somewhere other than the police station? Why can't the police officer be trusted? Why do the detectives' names conveniently begin with the first two letters of the alphabet? This is Puzzleton, where certain things just are, and should not be questioned... -KD] Source: Bert Sevenhant, citing Vladomir Mascharouk, from White Russia (Belarus).

Adjacent Sums
Place the numbers 1 thru 9 in a 3x3 grid such that every adjacent pair (excluding diagonals) has a different sum. Your first solution will not use three possible sums. Repeat #1, using these sums. What is the largest possible difference between two adjacent numbers in the grid? Can you find solutions with this difference to satisfy #1 and #2 above? Source: Original.

Queen's Quadrille
From a standard chess set, remove the pawns and the white queen. Place the remaining pieces on a 4x4 chessboard (leaving one empty space.) Pieces move as in regular chess, but moves don't need to alternate in color and no piece can be captured (removed). The object is to move the queen along a specified path. Find a configuration and moves to let the queen visit every square of the board in the minimum possible total moves. You may count the starting square as "visited". The theoretical minimum is 29 moves (the queen moves 15 times, each other piece once.) How close can you get to this minimum? Find a configuration and moves to let the queen visit every square and return to her starting square in the minimum number of moves (the theoretical minimum is 31 moves, as explained above.) Source: Original, based on a game described in the June 1998 Games magazine, Page 50, Karen Deal Robinson.

Deterministic Checkers
In checkers, can you find or create a position from which, if white moves correctly, the remaining moves must result in white's win? This doesn't necessarily mean that each player only has one possible move, just that any possible move leads to white's win. Try to maximize the number of remaining moves in the game. Try to maximize the number of pieces removed in the rest of the game. (This could be the same solution as the previous question.) Try to maximize the number of white pieces removed in the rest of the game. (Again, this could be the same.) To make solutions similar, assume the opposing color is red, as at the following site for the rules of checkers. In an unrelated question, what is the maximum number of checkers which can be left on the board at the end of a standard game? Summary of particular rules: All jumps must be taken. If multiple choices for jumps exist, any can be chosen, but all pieces in the jump must be taken (you can't stop in the middle of a multiple jump.) A checker being crowned in the King's row ends that move. The game is lost when you either can't move or lose all your checkers. Source: Original, based on this similar problem in the June 1998 Games magazine, Page 63, John R. Gibson

Puzzles
This is the continuation of the submitted puzzles supplied by Philippe Fondanaiche (Paris, France). Feel free to send partial solutions.
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?

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?

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.)

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

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

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

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