Prove that you can not cover the plane with infinite strips which have
Take a circle with radius 1 in the plane. A strip with width X covers at most an area of 2X of the circle. Therefore all strips cover at most an area of 2, which is smaller than the total area of the circle (~3.14).
Find a non-intersecting path between the two holes in this maze, which passes through each of the balls in the given order.
The solution is shown below.
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:
Show that this method shuffles the deck uniformly, i.e. every permutation has the same chance to appear.
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 chances are you have already seen the hundreds of optical illusions we have collected for you on Puzzle Prime, but have you ever encountered any audio illusions? The YouTube channel AsapSCIENCE has created a short video in which they present and explain some of the most famous audio illusions, such as the McGurk effect and the Tritone paradox. Watch their video below and see if you can trust your ears.
Can you construct a convex polyhedron, such that no two of its faces have the same number of edges?
No, you can not construct such a polyhedron. Assume the opposite and consider its face, which has the largest number of sides, say k. Then the polyhedron contains at least k more faces with different numbers of sides, all less than m. However, this is clearly impossible.
Someone stole gold coins from a museum near the park. No one saw the thief take the coins, so there isn’t a description of the robber. Slylock Fox suspects one of the creatures in the park is the thief. Which one?
The raccoon on the seesaw couldn’t hold the heavier bear off the ground unless he was carrying something heavy. Since gold is one of the heaviest metals, Slylock suspects the raccoon is the thief and has hidden the coins in his clothes.
In 1996, just a day before the election of the 40th President of US, the 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 a 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.
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 the squares above it except for h8 are covered by pawns.
Assign the following weights on the squares of the chessboard:
Every time you make the splitting move, the total sum of the numbers of the squares covered by pawns remains a constant. At 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.
If you know that the following game has been monochromatic, i.e. no piece has moved from black to white square or vice-versa, which one is the correct position of the bishop – e3 or e4?
The correct position of the bishop is e3. Otherwise, no White’s piece could have captured the last Black’s piece, moving on black squares.
There are several pieces of coal, a carrot, and a scarf lying on a lawn. If nobody placed them there, how did they end up like that?
Somebody made a snowman which melted within a few days.
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