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A Thread About Favourite Puzzles

Discussion in 'Mathematics' started by maths126, Jul 30, 2015.

  1. maths126

    maths126 New commenter

    Wondering if folk have particular favourites amongst the many puzzles in our repertoire. I'll start the ball rolling with the 12 coins puzzle, reproduced below for convenience.

    The 12 Coins Puzzle
    You have 12 coins, one of which is counterfeit. It is either heavier than the other 11 coins, or lighter than the others - you do not know. Using only three weighings on a two-pan balance scale, how would you determine which is the counterfeit coin?
  2. Owen134866

    Owen134866 New commenter

    This one is definitely one of my all time favourites, can still remember the feeling of the light coming on in my head when I worked out how to think about it :)

  3. lou1990lou

    lou1990lou New commenter

    I really like to do a good ol' kenken!
  4. maths126

    maths126 New commenter

    Among those authors whose puzzle collections are consistently good, I would list the following:

    Martin Gardner (Everything!)

    Raymond Smullyan (Logic puzzles, True-False or Knights & Knaves)

    Boris A. Kordemsky (The Moscow Puzzles)

    Paul Sloane (Lateral thinking puzzles)

    David Wells (Curious & Interesting Puzzles - a good 'classics' collection)

    Other recommendations?
  5. PFCDaz

    PFCDaz New commenter

  6. Colleen_Young

    Colleen_Young Occasional commenter

    I rather like those area puzzles - found them through Twitter the other day.

    More puzzles here.
  7. In a room full of students, prove that at least two students have the same number of friends as each other (presuming we're limiting friendships to those amongst students in the room).
  8. frustum

    frustum Star commenter

    For those who like pencil-and-paper puzzles, try www.ukpuzzles.org. If you fancy getting competitive, tomorrow is India's annual puzzle championship - you can compete online.
  9. fieldextension

    fieldextension New commenter

    Never seen that before. I take this there is nothing quicker than this?

    Suppose there are n people in the room. Then the most friends any one person can have is n-1. To avoid anyone having the same number of friends, the numbers of friends each person has would need to be n-1, n-2, n-3, all the way down to 0 friends for the last person. But if the last person has no friends then the first person cannot have n-1 friends. So having at least two people with the same number of friends cannot be avoided.

    It is vaguely reminiscent of this:

    A climber ascends a mountain, setting off at 9 am on Monday and reaching the summit later that day. He sleeps at the summit and descends the mountain, starting at 9am the next (i.e. Tuesday) morning. Prove that there exists at least one (identical) height reached at the same time of day on both days.
  10. Spot on! You're implicitly using the 'Pigeonhole Principle' at the end, the idea that because you have more students than possible numbers of friends, at least two students have the same number of friends.

    A harder one courtesy of Erdos: "Show that if you pick n+1 numbers from 1 to 2n, one must divide another".
  11. Consider a scalene triangle ABC (with A at the 'top'). Draw a line from B to somewhere on AC. Draw another line from C onto AB. You should find the triangle divided into three triangles and a quadrilateral at the top. If the area of bottom triangle is 10, the area of the left triangle is 8 and the area of the right triangle is 5, find the area of the quadrilateral.

    Took me ages to do this when I was at school.
  12. 22. Used the fact that the area of triangles with the same base is proportional to the height. Which incidentally is sort of now in our Year 7 syllabus: www.drfrostmaths.com/resource.php
  13. fieldextension

    fieldextension New commenter

    My personal favourite is Peter Winkler's ice cream cake puzzle. Link

    The website at the link, and Winkler's book, both come with my recommendation (I am not on commission).

    The ant problem mentioned first in the podcast is also a classic.

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