Round 274

Date Released: August 14th, 2008.
Date Rewarded: August 21st, 2008.

A Chomby was walking down a path one day when Jhudora appeared in front of it. "Solve this puzzle, and you may pass," she said.

Suppose you have a bunch of wooden beds, each weighing 200 pounds each, arranged in a pyramid. The base layer is 5 beds by 5 beds, the layer above is 4 beds by 4 beds, and so on until the top layer of one bed. They are arranged in such a way that each bed is resting on four beds below it, one leg on each bed.

If the weight on each bed is always equally distributed to all four legs, and each bed leg can only support 200 pounds, how many pounds can you place on top of the topmost bed? Please round to the nearest pound, and plese submit only a single number as your answer.

Prize


Wooden Bed

Click to show/hide solution

If the weight on each bed is always equally distributed to all four legs, and each bed leg can only support 200 pounds => Each bed can support maximum 200 x 4 = 800 pounds.

Each bed weighs 200 pounds. Let x be the extra weight you can put on the top of the topmost bed. We will go from top to bottom level.

Top level: total weight = x + 200. Each bed leg can withstands: (x + 200) / 4.

2nd top level: see the below figure for visual help, the four red dots are the four legs of the top level bed. The total weight of each bed (including itself) = (x + 200) / 4 + 200 = (x + 1000) /4. Each bed leg withstands: (x + 1000) / 16.


3rd level: see the below figure for visual help, the red dots are the 2nd top level beds' legs.

* middle bed: the bed withstands 4 legs of the above 4 beds (one of each). Therefore, the total weight should be: (x + 1000) / 16 * 4 + 200 = (x + 1800) / 4. Each bed leg withstands: (x + 1800) / 16.

* side beds: each bed withstands 2 legs of the above 2 beds (one of each). Therefore, the total weight should be: (x + 1000) / 16 * 2 + 200 = (x + 2600) / 8. Each bed leg withstands: (x + 2600) / 32.

* corner beds: each bed withstands 1 leg of the above 1 bed. Therefore, the total weight should be: (x + 1000) / 16 + 200 = (x + 4200) / 16. Each bed leg withstands: (x + 4200) / 64.


4th level: see the below figure for visual help, the red dots are the 3rd level middle bed's legs, the yellow dots are the 3rd level side beds' legs and the blue red dots are the 3rd level corner beds' legs.

Since we only concern about the bed that withstand the most weights, we can ignore the side and corner beds and only consider the 4 middle beds which contributes most weights into the middle bed on the bottom level (see the bottom level figure). Each of the 4 middle beds withstand 1 leg from a 3rd level corner bed, 2 legs from 3rd level side beds, 1 leg from the 3rd level middle bed. Therefore, the total weight of each 4th level middle bed should be: (x + 1800) / 16 + (x + 2600) / 32 * 2 + (x + 4200) / 64 + 200 = (9x + 34600) / 64. Each bed leg withstands: (9x + 34600) / 256.


Bottom level: see the below figure for visual help, the red dots are the 4th level middle beds' legs, the yellow dots are the 4th level side beds' legs and the blue red dots are the 4th level corner beds' legs.

Similar to the 4th level, we only concern with the bed that has the most weight on. That bed is the middle bed. Therefore, the weight of on that bed is: (9x + 34600) / 256 * 4 + 200 = (9x + 47400) / 64. Each bed leg withstands: (9x + 47400) / 256.


We know that each beg leg can only withstand maximum 200 pounds. From all the levels, the bed that has most weights on is the middle bed of the bottom level. Therefore:

(9x + 47400) / 256 <= 200 <=> x <= 3800/9 = 422.22

Therefore, the answer is 422.

Results: 1432 people guessed the correct answer earning themselves 1397 NP each.

Round 20

Date released: August 2nd, 2001.
Date rewarded: August 10th, 2001.

The economy of Roo Island is rather strange. They have four coins, each worth a different number of Neopoints. Three red coins plus two Neopoints equal a green coin. Three green coins plus five Neopoints equal a purple coin. Eleven red coins plus a single Neopoint equal a purple coin.

The fourth coin is a silver coin that is worth four purple coins. If I go to the Roo Island bakery and buy a loaf of bread worth 84 Neopoints with my silver coin, and the shopkeeper gives me change in red coins... how many red coins would I get?

Prize
Unknown

Click to show/hide solution

Let r, g, p, s be the values of red, green, purple and silver coins respectively. We know that:

* Three red coins plus two Neopoints equal a green coin:

3r + 2 = g (1)

* Three green coins plus five Neopoints equal a purple coin:

3g + 5 = p (2)

* Eleven red coins plus a single Neopoint equal a purple coin:

11r + 1 = p (3)

* Silver coin is worth four purple coins:

s = 4p (4)

From (2) and (3), we have:

3g + 5 = 11r + 1 <=> 11r = 3g + 4 (5)

From (1) and (5), we have:

11r = 3g + 4 = 3 * (3r + 2) + 4 = 9r + 10 <=> 2r = 10 <=> r = 5

Therefore:

g = 3r + 2 = 17
p = 3g + 5 = 56
s = 4p = 224

The change that I should get (in red coins):

change = (224 - 84) / 5 = 140 / 5 = 28

I should get back 28 red coins.

Round 16

Date released: July 6th, 2001.
Date rewarded: July 10th, 2001.

Senator Barca came rushing over the Sand Dune to talk to Peopatra. 'Help, Im being chased by Grundos. Can I hide here???'. Peopatra nodded and lifted up the carpet concealing the trapdoor in the floor of her tent.

All but two of the Grundos chasing Barca were Green, all but two of the Grundos were Purple, and all but two of the Grundos were Red.

How many Grundos were chasing Barca???

Prize
Unknown

Click to show/hide solution

Let x be the number of Grundo.
Number of Green Grundos: x - 2
Number of Purple Grundos: x -2
Number of Red Grundos: x -2

Therefore, number of all Grundos:

(x - 2) + (x - 2) + (x - 2) = x <=> 3x - 6 = x <=> 2x = 6 <=> x = 3

There are 3 Grundos. (one of each color)

Round 15

Date released: June 27th, 2001.
Date rewarded: July 6th, 2001.

Two kings of Sakhmet from history were King Galat-Ra the Lupe and King Hephaes-Ru the Kacheek. Galat-Ra died 120 years after Hephaes-Ru was born. Their combined ages when they died was 100 years. Hephaes-Ru died in the year 40 B.N. (Before Neopia).

In what year was King Galat-Ra born?

Prize


Geb

Click to show/hide solution

Let x be the year King Galat-Ra was born, y be the year King Hephaes-Ru was born. We know that: "Galat-Ra died 120 years after Hephaes-Ru was born". Therefore, Galat-Ra died in the year y + 120. His age when he died is y + 120 - x. Hephaes-Ru's age when he died is 40 B.N. - y.

We also know that: "Their combined ages when they died was 100 years." Therefore:

y + 120 - x + 40 B.N. - y = 100 <=> x = 40 B.N. + 20 = 20 B.N.

King Galat-Ra was born in 20 B.N.

Round 10

Date Released: May 23rd, 2001.
Date Rewarded: May 30rd, 2001.

Siya the Usuki Doll (pictured above) is 9 years older than her brother, and 25 years younger than her father. In seven years time her father will be three times older than her brother. How old is Siya now?

Prize
Unknown

Click to show/hide solution

Let x be Siya's age, y be her brother's age and z be her father's age. We know:

y = x - 9
z = x + 25

In seven years, her brother's age and her father's age will be x - 2 and x + 32 respectively (add 7 to y and z). Her father's age will be three times her brother's age then. Therefore,

x + 32 = 3 * (x - 2) <=> x + 32 = 3x - 6 <=> 2x = 38 <=> x = 19

Therefore, Siya is 19 years old now.

Check: her brother's age = 10, her father's age = 44. In seven years, her father's age = 51, three times as much as her brother's age = 17.

Note: The picture is missing.

Round 2

Date released: March 27th, 2001.
Date rewarded: April 3rd, 2001.

A Skeith weighs 100 pounds plus half of his weight. How much does he weigh?

Prize
Unknown

Click to show/hide solution

Let x be the weight of the Skeith. We have:

x = 100 + 0.5x <=> 0.5x = 100 <=> x = 200

Another way to look at the puzzle is: the skeith weighs 100 pounds plus half of his weight. Therefore, 100 pounds must be the other half of his weight. Then, his weight is 200 pounds.

Therefore, the Skeith weighs 200 pounds.

Note: the game has the trophies.