PASSING THE TRACK

Two cars (A and B) start at the starting line at the same time on a 3-mile long track, going in opposite directions. As they drive around the course, they pass each other many times. Exactly one hour after starting, they pass each other for the 33rd time. At this point car A has completed exactly 20 laps. What is the average speed of Car B?

Answer:

Suppose car B would stand still on the starting line. Then car A would pass car B 20 times in one hour. Every extra passing would be the result of the driving of car B. Because they pass 33 times, car B must have completed 13 laps giving a distance of 13 x 3 = 39 miles. [and a speed of 39 mph.]

A bridge teaser 1.

In one hand of bridge, all four deuces took tricks. Game was bid and made. How many overtricks?


None




In order for a deuce to take a trick, all other players must be void in that suit.


Consider the last time during the game that a non-trump deuce takes a trick.

1. No trump is played, or this deuce would not win the trick.


2. No two cards played in this hand can be the same suit. For if they are, that means that at least two players are not void in that suit and the deuce of that suit could not have taken a trick.




We have four diffent suits (from 2) and none of them are trump (from 1) so we must be playing in no trump.



Again, since in order for a deuce to take a trick, all other players must be void in that suit, we can deduce that each player must have won with the deuce of the suit that he played on this trick.



A deuce can only take a trick if it is on lead, and it cannot win opening lead, so the defenders must have won the lead twice in addition to winning with the two deuces. So four tricks went to the defense. Since gane was bid and made, that means that the winners must have taken all remaining trciks, exactly making the contract of 3 No Trump.

A BRIDGE TEASER

Bridge

You are walking past a bridge table when one of the players is called away to an urgent phone call.

Five cards are thrust into your hand, and you are told "You need all the remaininbg tricks, there is no trump."




You cash the three aces and both opponents follow suit. You then lead the club deuce and the next hand plays small. Do you play the Ace or the queen?




Answer:

---

Bridge

Play the Queen.

Let's assume that your remaining opponent has The King of clubs. That would mean that everybody had a club this hand.

Since everybody followed suit when you cashed the three Aces, this means that no one had a void when you came into the game

When no one has a void, that mean that everyone followed suit on all previous tricks, and thus the cards have come out in groups of four from the same suit. The number of cards of each suit left in the players hands must therefore be either 13, 9, or 5. When we came to the table, there must have been at least 5 spades, 5 hearts, 5 diamonds, and 9 clubs (five isn't enough, because we have four and we must allow at least one for each of our opponents). This makes a total of 24 cards. But since each player only has 5 cards when we start, this is impossible.

Our original assumption must have been wrong. East doesn't have the King of Clubs, and playing the queen will win both remaining tricks.
please see



here.





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It bugs me when students assume that a function f(x) and its Taylor series

are always the same.
They aren't, of course. A simple counter-example is Modulus x function

The Taylor series at 1 is 1+(x-1), with the rest of the terms equal to zero. But this sums to x, which does not equal f(x) for negative x's.