What is the peculiar property of this sequence of words?

WHAT, HOLE, ALES, TEST

SOLUTION

If you take the first letter from each word, you will spell the first word in the sequence. If you take the second letter from each word, you will spell the second word in the sequence. If you take the third letter from each word, you will spell the third word in the sequence. If you take the fourth letter from each word, you will spell the fourth word in the sequence.

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.

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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