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You have ten lanterns, five of which are working, and five of which are broken. You are allowed to choose any two lanterns and make a test which tells you whether there is a broken lantern among them or not. How many tests do you need until you find a working lantern?
Remark: If the test detects that there are broken lanterns, it does not tell you which ones and how many (one or two) they are.
You need 6 tests:
(1, 2) → (3, 4) → (5, 6) → (7, 8) → (7, 9) → (8, 9)
If at least one of these tests is positive, then you have found two working lanterns.
It all of these tests are negative, then lantern #10 must be working. Indeed, since at least one lantern in each of the pairs (1, 2), (3, 4), (5, 6) is not working. Therefore, there are at least 2 working lanterns among #7, #8, #9, #10. If #10 is not working, then at least one of the pairs (7, 8), (7, 9), or (8, 9) must yield a positive test, which is a contradiction.
With some case analysis, it is not hard to see that 5 tests are not enough.
The sentence below is grammatically correct. Can you explain it?
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
The sentence says that buffalo (animals) from Buffalo (city, US), which are buffaloed (intimidated) by Buffalo (city, US) buffalo (animals), themselves buffalo (intimidate) buffalo (animals) from Buffalo (city, US).
I build bridges of silver and crowns of gold. Who am I?
The answer is DENTIST.
A 1000 × 1004 rectangle is split into 1 × 1 squares. How many of these squares does the main diagonal of the large rectangle pass through?
Notice that the number of small squares the main diagonal passes through is equal to the number of horizontal and vertical lines it intersects. Indeed, every time the diagonal goes through the interior of one square to the interior of another, it must intersect one of these lines.
There are 1000 + 1004 = 2004 lines which are intersected by the main diagonal. However, on four occasions (which is the greatest common divisor of 1000 and 1004), the main diagonal intersects one horizontal and one vertical line at the same time, which results in double-counting., so we must subtract 4 from the answer.
Therefore, the answer is 1000 + 1004 – 4 = 2000.
Two twins are lying next to a king and a queen in a large room. Yet, there are no adults and there are no children in the room. How is it possible?
The twins, the king, and the queen, are all (types of) beds.
The sides of a rectangle have lengths which are odd numbers. The rectangle is split into smaller rectangles with sides which have integer lengths. Show that there is a small rectangle, such that all distances between its sides and the sides of the large rectangle have the same parity, i.e. they are all even or they are all odd.
Source: Shortlist IMO 2017
Split the large rectangle into small 1×1 squares and color it in black and white, chessboard-style, such that the four corner squares are black. Since the large rectangle has more black squares than white squares, one of the smaller rectangles also must have more black squares than white squares. Therefore, the four corners of that smaller rectangle are all black. Then, it is easy to see that all distances between its sides and the sides of the large rectangle have the same parity.
A man is sitting in his cabin in Colorado. Three hours later he gets out of his cabin in Texas. How is this possible?
The man is a pilot and he is sitting in his airplane cabin.