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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!
What was the challenge in escaping the tower?
Note that difference between weights in 2 baskets can't exceed 20 pounds. However, chest can withstand the shock even if the difference exceeds that limit
1.
They put 60 pounds chest in one of empty basket & sends it down. (60 pounds vs 0 Pounds)
2. Sosthenes of 80 pounds climbs in other empty basket and rides down to the ground while bringing up the chest. (60 pounds vs 80 pounds)
3. Now chest is taken out and Hilary is sent down while bringing up Sosthenes.(80 pounds vs 100 pounds).
4. Hilary gets out of the basket on ground and Sosthenes get out of the basket at top of the tower.
5. They again send chest of 60 pound to the ground putting it in 1 basket. This brings other empty basket up at the top of the tower.
6. Now, Hilary (who is on ground) climbs in the basket with chest (total weight = 60 + 100 = 160 pounds) while Babylas (170 pounds) enters in the basket at the top of tower. This allows Babylas to ride down to the ground. (160 pounds vs 170 pounds).
Right now, Babylas is on ground and Hilary, Sosthenes are on the top of the tower with chest.
7. Here, Babylas jumps on the ground & Hilary gets out of the basket at the top of tower.
8. Again, because of chest which weighs 60 pounds the basket descends down to the ground bringing up other basket up. (60 pounds vs 0 Pounds).
9. Now, Sosthenes (at top) rides down to the ground bringing up the chest. (60 pounds vs 80 Pounds).
10. Hilary replaces himself with the chest at the top and rides down while bringing Sosthenes up. (100 pounds vs 80 Pounds).
At the moment, Sosthenes is at the top with chest while Hilary and Babylas are on the ground.
11. Now, Sosthenes sends back chest on the ground & then rides to the ground while entering into other empty basket. (60 pounds vs 80 Pounds).
12. And when Sosthenes gets out of the basket on the ground the basket with chest descends on it's own to the ground.
13. Finally, all Sosthenes, Babylas and Hilary take chest out of the basket and safely escape the tower.
A motorcyclist was sent by the post office to meet a plane at the airport.
The plane landed ahead of schedule, and its mail was taken toward the
post office by horse. After half an hour the horseman met the
motorcyclist on the road and gave him the mail.
The motorcyclist returned to the post office 20 minutes earlier than he was expected.
How many minutes early did the plane land?
Here is calculation of scheduled arrival time.
