Shuffling Cards

52 cards - 2 of clubs to Ace of clubs, 2 of diamonds to Ace of diamonds, 2 of hearts to Ace of hearts, and 2 of spades to Ace of spades, are arranged in a deck. We shuffle them in the following manner:

  • We take the top card and put in a random place inside the deck
  • Once we get to the King of spades and put it somewhere in the deck, we stop

Show that this method shuffles the deck uniformly, i.e. every permutation has the same chance to appear.


Solution

Notice that at all times the cards below the King of spades are shuffled uniformly. Therefore at the end, after we put the King of spades in a random place inside the deck, the entire shuffle will be uniform as well.


The Coolest Crossword of All Time

In 1996, just a day before the election of the 40th President of US, New York Times published a curious crossword. In the 8th row, the solver should discover a phrase - the "lead story of tomorrow's newspaper". More precisely - the name of the future President of the country appears there. But how could New York Times know whether it was going to be Clinton or Bob Dole?


ACROSS:

1. "___ your name" (Mamas and Papas lyric)
6. Fell behind slightly
15. Euripides tragedy
16. Free
17. Forecast
19. Be bedridden
20. Journalist Stewart
21. Rosetta ???
22. 1960s espionage series
24. ___ Perigion
25. Qulting party
26. "Drying out" program
28. Umpire's call
30. Tease
34. Tease
36. Standard
38. "The Tell-Tale Heart" writer
39. Lead story in tomorrow's newspaper, with 43A
43. See 39A
45. Gold: Prefix
46. ___ Lee cakes
48. Bobble the ball
49. Spanish aunts
51. Obi
53. Bravery
57. Small island
59. Daddies
61. Theda of 1917's "Cleopatra"
62. Employee motivator
65. Otherworldly
67. Treasure hunter's aid
68. Title for 39A next year
71. Exclusion from social events
72. Fab Four name
73. They may get tied up in knots
74. Begin, as a maze

DOWN:

1. Disable
2. Cherry-colored
3. Newspaperman Ochs
4. Easel part
5. Actress Turner
6. Ropes, as dogies
7. Place to put your feet up
8. Underskirt
9. First of three-in-a-row
10. Lower in public estimation
11. Onetime bowling alley employee
12. Threesome
13. English prince's school
14. '60s TV talk-show host Joe
18. Superannuated
23. Sewing shop purchase
25. TV's Uncle Miltie
27. Short writings
29. Opponent
31. Likely
32. Actress Caldwell
33. End of the English alphabet
35. Trumpet
37. Ex-host Griffin
39. Black Halloween animal
40. French 101 word
41. Provider of support, for short
42. Much debated political inits
44. Sourpuss
47. Malign
50. "La Nausee" novelist
52. Sheiks' cliques
54. Bemoan
55. Popsicle color
56. Bird of prey
58. 10 on a scale of 1 to 10
60. Family girl
62. Famous ___
63. Something to make on one's birthday
64. Regarding
65. Quite a story
66. Dublin's land
69. ___ Victor
70. Hullabaloo


The answer is simple, yet very impressive. The crossword's author, the mathematics professor Jeremiah Farrell, created the puzzle so that it could be solved in two different ways, revealing either "Clinton Elected" or "Bob Dole Elected" in the middle row. Many of the newspaper's readers didn't realize the prank and assumed New York Times was displaying bias towards one of the candidates. They started sending lots of angry letters and calling the editor, complaining about arguably the coolest crossword of all time.

1996 New York Times Presidential crossword

Pawns on the Chessboard

Six pawns are placed in the middle squares of the main diagonal of a chess board - b7, c6, d5, e4, f3, g2. You are allowed to take any pawn on the chessboard and replace it with two pawns - one on the square above it and one on the square on its right, in case there are empty squares there. If after several moves there are no more pawns on the main diagonal, show that all squares above it except for h8 are covered by pawns.


Solution

Assign the following weights on the squares of the chessboard:

  • 1 on the main diagonal a8 - h1
  • 1/2 on the diagonal b8 - h2
  • 1/4 on the diagonal c8 - h3
  • 1/8 on the diagonal d8 - h4
  • 1/16 on the diagonal e8 - h5
  • 1/32 on the diagonal f8 - h6
  • 1/64 on the diagonal g8 - h7

Every time you make the splitting move, the total sum of the numbers of the squares covered by pawns remains a constant. In the beginning that sum is 6. Since 7/2 + 6/4 + 5/8 + 4/16+ 3/32 + 2/64 = 6, all 27 squares above the main diagonal, except the top-right corner (on which you can not place a pawn in any way), must be covered by pawns at the end.


Game of Chess

Ned and Jon are playing chess. Eventually, they end up in a position in which Ned (whites) is left with 2 rooks, and Jon (blacks) has just his king on the board. If Ned can mate Jon in exactly 4 different ways, what is the position of the pieces?


Solution

Black king on a1, white king on e5, white rooks on c2 and h1. Ned hasn't moved his king and rook, so he can either castle or move his king to d2, e2 or f2, resulting in a mate.


Deez Nuts

While changing a tyre, a motorist accidentally dropped all four wheel nuts into the sewer grate. Just when the man lost all hope to retrieve the nuts and continue on his way, a kid passed by. After hearing the story, the kid gave an advice, which enabled the driver to successfully fit a new tyre and drive to the nearest service station without any more accidents. What was the advice the kid gave to the motorist?


Solution

The kid suggested that the man uses one bolt from each of the other three wheels to fix the fourth one.


Euclidea

Which was your favorite part of the Mathematics you learnt back in high-school? Not sure about you, but we definitely loved the construction problems, where we had to draw some shape, using just a straightedge and a compass. Even though we haven't solved such problems for many years, we got very excited to discover the amazing game Euclidea, designed by the guys from HORIS International Ltd. If the description below seems intriguing to you, make sure to visit the game's website and test your skills by clicking the provided link.

Euclidea is all about building geometric constructions using straightedge and compass. About doing it the fun way. With Euclidea you don’t need to think about cleanness or accuracy of your drawing — Euclidea will do it for you. But it’s also a game. A game that values simplicity and mathematical beauty. Find the most elegant solution — the one, which is built in the least possible moves, — and you’ll get the highest score.

http://www.euclidea.xyz

The Lion and the Zebras

The lion plays a deadly game against a group of 100 zebras that takes place in the steppe (an infinite plane). The lion starts in the origin with coordinates (0,0), while the 100 zebras may arbitrarily pick their 100 starting positions. The the lion and the group of zebras move alternately:

  • In a lion move, the lion moves from its current position to a position at most 100 meters away.
  • In a zebra move, one of the 100 zebras moves from its current position to a position at most 100 meters away.
  • The lion wins the game as soon as he manages to catch one of the zebras.

Will the lion always win the game after a finite number of moves? Or is there a strategy for the zebras that helps them to survive forever?


Solution

The zebras can survive forever. They choose 100 parallel strips with width 300m each, then start on points on their mid-lines. If the lion lands on some zebra's strip, the zebra simply jumps 100m away from the lion, along its mid-line.