Mick, Nick, and Rick arrange a three-person gun duel. Mick hits his target 1 out of every 3 times, Nick hits his target 2 out of every 3 times, and Rick hits his target every time. If the three are taking turns shooting at each other, with Mick starting first and Nick second, what should be Mick’s strategy?
SOLUTION
Clearly, Mick should not aim for Nick, because if he kills him, then he will be killed by Rick. Similarly, Nick should not aim for Mick, because if he kills him, then he also will be killed by Rick. Therefore, if Rick ends up against alive Mick and Nick, he will aim at Nick, because he would prefer to face off a weaker opponent afterward. This means that if Nick is alive after Mick shoots, he will shoot at Rick.
Thus, if Mick shoots at Rick and kills him, then he will have to face off Nick with chance of survival less than 1/3. Instead, if he decides to shoot in the air, then he will face off Nick or Rick with chance of survival at least 1/3. Therefore, Mick’s strategy is to keep shooting in the air, until he ends up alone against one of his opponents.
Puzzle Prime’s third puzzle tournament was organized on January 31, 2021. Congrats to Elyot G. who is once again a winner! You can see the problems and the rankings below.
Elyot G.
You have 60 minutes to solve 5 puzzles, each worth 1 point. Upload your solutions as a pdf, document, or image, using the form below. Good luck!
Connect the stars with lines, so that the number inside each star corresponds to the number of lines connected to it, and the number in each barrier corresponds to the number of lines intersecting it.
Note: The stars in the corners cannot be connected, since the lines would pass through other stars.
PLAYING THE GAME: Draw lines to move Friends. A move is connecting a point on the Grid to another point on the Grid by drawing a straight line. The line can be made in any direction, as long as it connects exactly two points and doesn’t cross or touch another line.
TIMELINE: You move each Friend as many times as it says on the Timeline, starting with the topmost Friend and continuing in the order all the way down to the bottom one.
KEY POINTS, EXIT POINT, AND WALLS: One of the Friends must move to the Exit Point, but before he does that, each of the Key Points must be already connected with a line. A Wall is a double line and no Friend can cross or touch it.
Circles are particles and lines joining them are bonds. The objective is to find all the hidden values, following these four rules:
Particle values must be the sum of their bond values.
Particles can have the following values: 0, 1, 2, 4, 8, 12, 16.
Bonds can have the following values: 0, 1, 2, 4.
If two particles have the same value, the bond between them must have value 0.
SHOW EXAMPLE
PUZZLE
SOLUTION
4. Chess Connect
by Puzzle Prime
The starting and ending positions of 6 chess pieces are shown on the board. Find the trajectories of the pieces, if you know that they do not overlap and completely cover the board.
Notes: The pieces can not backtrack. Two trajectories can intersect diagonally but can not pass through the same square. Only the Knight has a discontinuous trajectory.
In the three murder cases below, you can read the testimonies of all suspects. For each case, find who the killer is, knowing that no 2 people are in the same row or column, and that the killer was alone in a room with the victim.
How many numbers between 1 and 100 can you pick at most, so that none of them divide another?
SOLUTION
You can choose fifty numbers at most: 51, 52, 53, … , 100.
In order to see that you cannot choose more than fifty, express each number in the form 2ⁿ×m, where m is an odd number. Since no two numbers can have the same m in their expressions, and there are only fifty odd numbers between 1 and 100, the statement of the problem follows.
There are 5 points in a square 1×1. Show that 2 of the points are within distance 0.75.
SOLUTION
Split the unit square into 4 small squares with side lengths 0.5. At least one of these squares will contain 2 of the points. Since the diagonals of the small squares have lengths less than 0.75, these 2 points must be within such distance.
Take a square (or circle) coaster. Then, cut a hole in a piece of paper with the shape of a square, so that the side of that square is half of the side (or the diameter) of the coaster. Now, your goal is to push the coaster through the hole without tearing the paper (you can fold it).
Is it possible to connect each of the houses with the well, the barn, and the mill, so that no two connections intersect each other?
SOLUTION
No, it is impossible. Here is a convincing, albeit a informal proof.
Imagine the problem is solvable. Then you can connect House A to the Well, then the Well to House B, then House B to the Barn, then the Barn to House C, then House C to the Mill, and finally the Mill to House A. Thus, you will create one loop with 6 points on it, such that houses and non-houses are alternating along the loop. Now, you must connect Point 1 with Point 4, Point 2 with Point 5, Point 3 with Point 6, such that the three curves do not intersect each other. However, you can see that you can draw no more than one such curve neither on the inside, nor the outside of the loop. Therefore, the task is indeed impossible.
More rigorous, mathematical proof can be made using Euler’s formula for planar graphs. We have that F + V – E = 2, where F is the number of faces, V is the number of vertices, and E is the number of edges in the planar graph. We have V = 6 and E = 9, and therefore F = 5. Since no 2 houses or 2 non-houses can be connected with each other, every face in this graph must have at least 4 sides (edges). Therefore, the total number of sides of all faces must be at least 20. However, this is impossible, since every edge is counted twice as a side and 20/2 > 9.
Kuku and Pipi decide to play a game. They arrange 50 coins in a line on the table, with various nominations. Then, alternating, each player takes on their turn one of the two coins at the ends of the line and keeps it. Kuku and Pipi continue doing this, until after the 50th move all coins are taken. Prove that whoever starts first can always collect coins with at least as much value as their opponent.
Remark: On the first turn, Kuku can pick either coin #1 or coin #50. If Kuku picks coin #1, then Pipi can pick on her turn either coin #2 or coin #50. If Kuku picks coin #50, then Pipi can pick on her turn either coin #1 or coin #49.
SOLUTION
Let’s assume Kuku starts first. In the beginning, he calculates the total value of the coins placed on odd positions in the line and compares it with the total value of the coins placed on even positions in the line. If the former has a bigger total value, then on every turn he takes the end coin which was placed on odd position initially. If the latter has bigger value, then on every turn he takes the end coin which was placed on even position initially. It is easy to see that he can always do this because after each of Pipi’s turns there will be one “odd” coin and one “even” coin at the ends of the line.
Cheryl tells the month to Albert and the day to Bernard.
Albert: I don’t know the birthday, but I know Bernard doesn’t know either. Bernard: I didn’t know at first, but now I do know. Albert: Now I also know Cheryl’s birthday.
When is Cheryl’s birthday?
SOLUTION
If Albert knows that Bernard doesn’t know when the birthday is, then the birthday can’t be on May 19 or June 18. Also, Albert must know that the birthday can’t be on these dates, so May and June are completely ruled out.
If Bernard can deduce when the birthday is after Albert’s comment, then the birthday can’t be on 14th. The remaining possibilities are July 16, August 15, and August 17.
Finally, if Albert figures out when the birthday is after Bernard’s comment, then the date must be July 16.
Puzzle Prime’s second puzzle tournament was organized on June 27, 2020. Congrats to Elyot G. who solved all the puzzles! You can see the problems and the rankings below.
Elyot G.
You have 60 minutes to solve 6 puzzles, each worth 1 point. Upload your solutions as a pdf, document, or image, using the form below. Good luck!
Time for work: 1 hour
Puzzle Prime Knight
Start from a square with a P on the chessboard, and keep jumping via knight’s move, consequently landing on squares with the letters U-Z-Z-L-E-P-R-I-M-E.
Point of View
8 of these diagrams correspond to views of the object in the corner when it is looked from different perspectives. Which 2 aren’t?
Note: The projections below are parallel (not perspective).
Chess Fight
Choose a chess piece on the board. Then, move the piece to a cell with another piece, and remove the first piece. Repeat, by moving the second piece to a cell with a new piece, and removing it. Continue until there is only one piece remaining on the board.
Note: For example, we can move the Queen on d2 to the Bishop on b2, the Knight on c3, or the Bishop on d4. If we move the Queen to d4 and remove it from the board, then we must move the Bishop on d4. The only available cell is c3, where a Knight is positioned. We must remove the bishop and move the Knight on c3 either to the Rook on b1 or the Rook on a2…
Special Date
On April 5, 2013 (5.4.2013), the digits used for expressing the date were all different and consecutive. When was the last date before it with this property?
Remark: The digits 9 and 0 are not consecutive.
Splitting the Area
You have 1 square with side length 1 and 2 circles with diameter length 1. Draw a single line so that the resulting areas on the left and on the right of the line are equal.
Notes: You need to specify how you find a line satisfying the condition above.
Chess Connect
The starting and ending positions of 6 chess pieces are shown on the board. Find the trajectories of the pieces, if you know that they do not overlap and completely cover the board.
Notes: The pieces can not backtrack. Two trajectories can intersect diagonally but can not pass through the same square. Only the Knight has a discontinuous trajectory.
Solutions
The answer to Point of View is C and J. The answer to Special Date is 23.4.1765. The solutions of the other puzzles are shown below.
Puzzle Prime KnightChess FightSplit the AreaChess Connect
