Hidden Objects Christmas
Can you find the objects hidden in the three pictures below? You are looking for:
- 1 sleigh in the traffic
- 1 elf in the traffic
- Santa Claus in the traffic
SOLUTION
The solution is shown below.
Category: Puzzles
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Brick in the Wall
On the picture, you can see an example of a wall made of 2×1 bricks. On the wall, there are 2 cracks, which are straight lines passing through the whole wall from top to bottom and from left to right, without intersecting any bricks.
Can you make the following walls without any cracks:
- wall 5×6 with 15 bricks;
- wall 6×6 with 18 bricks?
SOLUTION
The solution for a 5×6 wall is shown below. However, if the wall has dimensions 6×6, it is impossible to build it without any cracks. Indeed, assume the wall does not have any cracks. Therefore every line passing through it must intersect 2, 4, or 6 bricks. Since there are in total 10 lines passing through the wall and each brick is intersected by exactly one of them, the total number of bricks must be at least 10 x 2 = 20 > 18. This yields a contradiction.
Integer Dimensions
A large rectangle is partitioned into smaller rectangles, each of which has integer length or integer width. Prove that the large rectangle also has integer length or integer width.
SOLUTION
This problem can be solved using graph theory, but the most elegant solution is based on some basic calculus.
Place the big rectangle in the plane so that its sides are parallel to the X and Y axes. Now integrate the function f(x)=sin(πx)sin(πy) over the boundary of any small rectangle. Since at least one of its sides has integer length, the result will be 0. If you sum all integrals taken over the boundaries of the small rectangles and cancel the opposite terms, you will get that the integral of f(x) over the boundary of the large rectangle is also equal to 0. Therefore at least one of its sides has integer length.
Meg’s Height
Four years ago, Meg put a nail on a tree in order to mark her height. If the tree grows 10 inches per year, and currently the nail is 5 inches lower than Meg, how much has Meg grown over these four years?
SOLUTION
Five inches. Trees grow at their tops.
Spades and Bridge
You are playing a game of Bridge. Which probability is greater – that you and your partner do not have any spades, or that you and your partner have all the spades in the game?
SOLUTION
You and your partner do not have any spades if and only if your opponents have all the spades. Therefore the probabilities are equal.
City of Stairs
Help the wanderer find his way to the lady.
SOLUTION
The solution is shown below.
The Silent Parrot
A person enters a pet shop and sees a beautiful parrot. The seller guarantees him that the bird repeats everything it hears, so the person buys it. However, when he goes back home and tests the bird, it turns out that it doesn’t say a word. The buyer goes to the shop to complain to the seller, but the seller argues he has not lied. How is this possible?
SOLUTION
The parrot is deaf.
The Sheikh and the Camels
A Sheikh dies, leaving behind three sons, 17 camels, and the following order:
1. The oldest son shall inherit one in two camels.
2. The middle son shall inherit one in three camels.
3. The youngest son shall inherit one in nine camels.
Now the three sons do not know what to do. They ask an old friend of the family for advice, and he finds a solution. What does the friend propose?
SOLUTION
The friend lent them one camel to make the camels 18. The first son took 9 camels, the second son took 6 camels, the third son took 2 camels. Then the friend took back his camel.