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.
You have four metal chains and each of them has three links. What is the minimal number of cuts you need to make so that you can connect the chains into one loop with twelve links?
SOLUTION
You need only three cuts – cut all the links of one of the chains and then use them to connect the ends of the remaining three chains.
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.
Someone tells you: “I’ll bet you $1 that if you give me $3, I will give you $5 in return”. Is this a good bet?
SOLUTION
You should not accept the bet and give him $3. If the person gives you the $5, then you would have given him $1+$3=$4, and you would make a profit of $1. However, the person can simply not give you anything and you will lose $3-$1=$2.
At some point in Leonel Messi’s career, the football player had less than 80% success when performing penalty kicks. Later in his career, he had more than 80% success when performing penalty kicks. Show that there was a moment in Leonel Messi’s career when he had exactly 80% success when performing penalty kicks.
SOLUTION
Let us see that it is impossible for Messi to jump from under 80% success rate to over 80% success rate in just one attempt. Indeed, if Messi’s success rate was below 80% after N attempts, then he scored at most 4N/5 – 1/5 = (4N-1)/5 times. If his success rate was above 80% after N+1 attempts, then he scored at least 4(N+1)/5 + 1/5 = (4N-1)/5 + 6/5 times. However, Messi can not score more than one goal in a single attempt, which completes the proof.
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