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A man stuck in a small sailboat on a perfectly calm lake throws a stone overboard. It sinks to the bottom of the lake.
When the water again settles to a perfect calm, is the water level in the lake higher, lower, or in the same place compared to where it was before the stone was cast in?
Did you think it will rise?
But why water level was affected?
Do you recall what does Archimedes Principle state? For an object to float on water, it has to displace that much volume of water whose weight is equal to weight of the object itself. Now if object has less density than water then obviously it has to displace lesser amount of volume of water to float on it. That means it has to sink less in water.
For a moment, let's assume the stone has very high density & hence having weight equal to hundreds of kilograms despite of having small volume.
Here, stone sinks to the bottom of the lake suggests that it is has more density than water. It can't displace the water whose weight is equal to it's weight.But when it was on sailboat it could push the sailboat down so that more water is displaced weighing equal to it's weight. Result of this, the sailboat sinks little 'deeper' compared to when stone wasn't there.
Obviously, the volume of displaced water when stone was in sailboat (due to stone only) must be greater that the volume of displaced water when stone sinks to the bottom of the lake. That's why both sailboat and stone together could float on the water. In short, sailboat helped stone to displace amount of water needed to float which results in rise in shoreline.
And when stone is thrown out of the sailboat, then ideally it can't displace more water than when it was on sailboat. Now, sailboat sinks less 'deeper' in water displacing only water need to float itself.
That's why the water level must be dropped compared to earlier. The little rise due to water displaced by stone can't exceed the earlier water level for the reason explained above.
A square tabletop measures 3n × 3n. Each unit square
is either red or blue. Each red square that doesn’t lie at the edge of
the table has exactly five blue squares among its eight neighbors. Each
blue square that doesn’t lie at the edge of the table has exactly four
red squares among its eight neighbors. How many squares of each color
make up the tabletop?
Here is correct way to count those!
How squares are arranged?
The tabletop measures 3n × 3n, so we can divide it evenly into n2 ( 3 × 3) squares that together tile the surface completely.
Let's consider a piece of square of size 3 x 3. For each such unit of 3 x 3 -
1. If the center of the square is red square, then there are 5 blue squares and 3 red squares surrounded with it.
2. If the center is blue square, then there are 4 blue and 4 red squares surrounding that square.
In any case, for 3 x 3 = 9 squares, there are 5 blue and 4 red squares.
Therefore, for tabletop of 3n x 3n, there will be 5n2 blue squares and 4n2 red squares.
One fine day, Mr. Puzzle and Mr. Fry were playing cards, but suddenly
power went off and they were getting bored. So Mr. Puzzle randomly
inverted position of 15 cards out of 52 cards(and shuffled it) and asked
Mr. Fry to divide the card in two pile with equal number of inverted
cards (number of cards in each pile need not be equal).
It was very dark in the room and Mr. Fry could not see the cards,
after thinking a bit Mr. Fry divided the cards in two piles and quite
surprisingly on counting number of inverted cards in both the piles were
equal.
What do you think Mr. Fry must have done?
This is what he must have done!
What was the challenge?
Mr.Fry must have taken top 15 cards & inverted positions of all. So he divides deck into 2 piles - one with 15 cards & other 37 cards.
Now suppose if there are 7 cards that were inverted in top 15 & 8 were inverted in remaining 37. When he flips top 15, 7 remains in normal position & 8 remains in inverted position. That is equal to 8 cards in inverted position from pile of 37 cards.
In short, if there are N cards inverted in top 15 then there are 15 - N cards inverted in remaining 37 cards. So on flipping position of top 15, there will be 15 - N cards in inverted position in top 15. That's how both piles would have equal number of inverted cards i.e. 15-N.
Had Mr.Puzzle inverted positions of 20 cards randomly then Mr. Fry would have flipped top 20 cards. He would have made 2 piles with one with 20 cards & other 32 cards to equate the count of inverted cards in piles.
Three robbers, Babylas, Hilary, and Sosthenes, are stealing a
treasure chest from the top of an old tower. Unfortunately, they’ve had
to destroy their ladder to avoid pursuit, so they’ll have to descend
using a crude tackle — a single pulley and a long rope with a basket at
each end.
Babylas weighs 170 pounds, Hilary 100 pounds, Sosthenes
80 pounds, and the treasure 60 pounds. If the difference in weight
between the two baskets is greater than 20 pounds then the heavier
basket will descend too quickly and injure its occupant (though the
treasure chest can withstand this).
How can the three of them safely
escape the tower with the treasure?
That's how they escape!