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TRICK PUZZLES (and some not-trick ones) – PART 3
(I am now running seriously thin on novel ideas…)

First, I present some (more) mathematical puzzles:

13) In a magic trick, a magician takes a standard set of double-six dominoes and removes the doubles. He asks a member of the audience to remove one domino (ideally at random), and arrange the other 20 to form a chain, all without the magician watching. The magician now looks at the chain, and instantly identifies the domino that was removed.

How does the trick work? And why does the magician insist that the doubles are removed first?

14) (Based on #89246) A book is opened to some arbitrary page. Unfortunately, some of the pages in the middle have been removed, so the two page numbers are not consecutive. Is it possible that the page numbers of the two shown are n and n² for some integer n?

15) The following ‘proof’ claims that 1=2, which is obviously wrong. Find the error.

Let x be a number satisfying x-1=2x-2. Then:
1/(x-1) = 2/(2(x-1)) = 2/(2x-2) = 2/(x-1)
Multiplying by (x-1) yields 1=2, as desired.

16) Let ABCD be a rectangle. Let M,N,P,Q be the midpoints of sides AB,BC,CD,DA respectively. Show that lines AC,BD,MP,NQ concur – that is, show that they meet at a single point. (Challenge: Show that this remains true even if ABCD is a parallelogram.)

Now some crosswords!

17) This crossword contains six words but only two clues… Gold luck!

(Here ? means a white square to be filled in; _ means a black square to NOT be filled in.)

Messages. (5)
Bowl with holes? (5)

18) All clues fit into a 3×3 grid with no gaps.

Clues (Across):
1. (e.g.) Black or green.
2. Anagram of Greek letter.
3. Past tense verb.

Clues (Down):
1. Anagram of 3 Down.
2. What you might do at lunchtime.
3. Sounds like number.

19) All clues fit into a 4×4 grid with no gaps.

Clues (Across):
1. ROT-13 cipher applied to 2 Across.
2. Can be made from the letters in BANANA.
3. Girl’s name.
4. Anagram of 3 Across.

Clues (Down):
1. Word with double letter.
2. Name of two of Henry VIII’s wives, with one letter changed.
3. Palindrome.
4. Bread.

The final puzzle here is definitely not a trick puzzle, but I have put it here because of how interesting it is. Enjoy!

20) After engaging in some dodgy actions (we will not disclose them here…!), you have been forced into playing a game with the Devil.

Here is how it works. The Devil writes on a blackboard the numbers 1 through 100. Beginning with the Devil, each player in turn colours one of the 100 numbers red, as long as it was previously uncoloured and it is either a factor or a multiple of the previous number to be coloured (if one exists). The first player that is unable to make a move wins.

For instance, a sample game could go:
41 → 82 → 1 → 86 → 2 → 94 → 47
And now the second player wins, since they are unable to make a move (both 1 and 94 have already been coloured).

Now, it is not too hard to see that the Devil will (comfortably) win with best play. (Challenge: Prove this.) However, in an act of religious miracle, a heavenly angel comes in and orders the Devil to begin with the number 13, giving you an advantage.

Can you now beat the Devil? If so, how do you respond?

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