Loading...

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.

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.

Loading...

Mick, Rick, and Nick arrange a three-person gun duel. Mick hits his target 1 out of every 3 times, Rick hits his target 2 out of every 3 times, and Nick hits his target every time. If the three are taking turns shooting at each other, with Mick starting first and Rick second, what should be Mick’s strategy?

Clearly, Mick should not aim for Rick, because if he kills him, then he will be killed by Nick. Similarly, Nick should not aim for Mick, because if he kills him, then he also will be killed by Nick. Therefore, if Nick ends up against alive Mick and Rick, he will aim at Rick, because would prefer to face off a weaker opponent afterward. This means that if Rick is alive after Mick shoots, he will shoot at Nick.

Now if Mick shoots at Nick and kills him, then he will have to face off Rick with chance of survival less than 1/3. Instead, if he decides to shoot in the air, then he will face off Rick or Nick 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.

Loading...

White plays and mates Black in one move. However, there is a mystery in this position which has to be revealed first.

SOLUTION

The mystery is that someone has just placed one extra black pawn on the board – there are 9 in total. Also, no matter which one is the added pawn, there always exists a mate in one move.

If the extra pawn was a7 – Qb6
If the extra pawn was b7 – Kc6
If the extra pawn was c4 – Qb4
If the extra pawn was d3 – Qe4
If the extra pawn was e3 – Bxf2
If the extra pawn was f7 – Ke6
If the extra pawn was g6 – Rg4
If the extra pawn was h3 – Rh4