Mathematical & Logical Puzzles -A1

1.       The Flight Around The World

                Three aeroplanes are sufficient to ensure the flight of one of them around the world.  There are many ways in which this may be done, but the following seems to be the most efficient.

                Planes A and B and C take off together.  After going one-eighth of the way, C transfers ¼ tank to A and ¼ tank to B.  This gives A and B full tanks and leaves C with a ¼ tank sufficient just to get back to base.

                Planes A and B continue a further one-eighth of the way when B transfers ¼ tank to A.  B now has ½ tank left, sufficient just to get back to base.

                Plane A meanwhile, with a full tank continues until he runs out of fuel ¼ way from base (¾ way round the world).  Here A is met by C which has been refuelled at base.  C transfers ¼ tank to A and both planes head for home.

                A and C run out of fuel one-eighth way from base where they are met by refuelled plane B.  B transfers ¼ tank to A and ¼ tank to C and the three planes then have just enough fuel to reach base with empty tanks.

2.       Who Is The Engineer?

·       From 2 we see that Mr. Robinson lives in Los Angeles.

·       From 3 and 6 we can deduce that the distinguished mathematical physicist lives in Omaha.

·       From 4 we see that Mr. Jones cannot live in Omaha (6) or Los Angeles (2) so he must live in Chicago.

·       From 5 we now know that the brakeman’s name is Jones.

From Smith cannot be the fireman; he cannot be the brakeman (5) so he must be the engineer.

3.       The Coloured Hats

C asks himself, “Can my hat be green?”  If so then A will know immediately that he has a red hat, for only a red hat on his head would cause B to lift his hand.  A would have therefore left the room. B would reason in the same way and also leave.  Since neither has left the room, C deduces that his own hat must be red.

4.       The Coloured Hats Again

If Bill and Colin are wearing red hats, Arthur would have known his hat was blue, because there were only two red hats.  Since Arthur didn’t know the correct answer, Colin concluded that there remained only two possiblities for himself and Bill.  Either both had blue hats, or one had a blue hat and the other a red hat.  If he himself had a red hat, Colin reasoned, Bill would have concluded that he, Bill, had a blue hat, because otherwise Arthur would have known that he, Arthur, must have a blue hat.  So Bill, because he was not able to tell the colour of his own hat, involuntarily betrayed to Colin that his, Colin’s hat was not red.  Therefore Colin concluded that his hat was blue.


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