Lenny Conundrum Round 396

In honor of the Daily Dare, AAA and Abigail decided to build themselves a tower.

The base of the tower is in the shape of a regular triacontakaihenagon, with each side measuring one metre on the exterior, and the tower rises vertically to a height of 31 metres, and it has a flat roof. The walls and roof are 30 centimetres thick.

If they decide to fill the tower with water, how many litres of water are required to fill the entire tower? Round to the nearest whole litre, and submit only a number for your answer; any extra information may disqualify your entry!

Possible Methods:

Answer:

 Lenny Conundrum Round 393

What is the next number in this sequence?

675
850
58
710
...

Possible Methods:

The numbers were derived from the difference between AAA and Abigail score for odd days of Daily Dare 2011.

Day 1: 1000 - 325 = 675

Day 3: 1000 - 150 = 850

Day 5: 70 - 12 = 58

Day 7: 800 - 90 = 710

Day 9: 32500 - 20000 = 12500

Answers: 12500

 Lenny Conundrum Round 389

A faerie gave a codestone to three lucky pets. The next day, those three pets each gave a codestone to another pet. The next day, those pets gave codestones to four other pets. The next day, those pets each gave a codestone to one other pet. The next day, those pets gave codestones to five other pets. And so on, in a fairly recognizable pattern.

Assuming each pet in Neopia received only one codestone, how many days would it take for every pet in Neopia to receive a codestone? Please round up to the nearest full day, and submit only the number as your answer.

Possible Methods:

Total number of pets = 270,138,851

The number of codestones given out for nth day, a_n is calculated using the formula "a_n = a_n-1 * n" where S_n-1 is the number of the codestones given out in the (n-1)th day and n is the number of codestones each pet gives out. n follows the pi sequence of 3.1415926535897.

The results can be computed as follows:

Answer: 14

Lenny Conundrum Round 387 

First, find the greatest integer that can divide each of the numbers 283, 427 and 715 to leave the same remainder in each case. Then, take that number, multiply it by four, and subtract 11. That number is X.

If you take the number you get by placing the digit 7 to the right of number Y, it is equal to Y + X.

What do you get when you multiply X by Y?

Possible Methods:

Let N be a common divisor and R be the common remainder.

Let A, B, C be quotient of 283, 427 and 715 respectively.

283 = AN + R  (1)
427 = BN + R  (2)
715 = CN + R  (3)

(2) - (1), 144 = (B-A)N (4)

(3) - (2), 288 = (C-B)N (5)

Greatest N must be the greatest common divisor of 144 and 288 based on the equations (4) and (5).

Since 288/144=2 and 144/144=1, 144 is the greatest integer that can divide each of the numbers 283, 427 and 715 to leave the same remainder in each case.

X = 144*4-11 = 565

X+Y = Y*10+7

9Y = 558

Y = 62

X*Y = 565*62 = 35030

Answer: 35030

 Lenny Conundrum Round 384

Once Gnorbus have 5 pounds of wool on them, they must be sheared. They grow wool at a rate of two pounds per month in the month of Celebrating, the month of Sleeping, and the month of Awakening. They grow wool at a rate of 1.2 pounds per month in the month of Running, the month of Eating, and the month of Hunting. They grow wool at a rate of 0.6 pounds per month in the month of Relaxing, the month of Swimming, and the month of Hiding. And they grow wool at a rate of 1.5 pounds per month in the month of Gathering, the month of Collecting, and the month of Storing.

Assuming they start the new year freshly sheared, and they can only be sheared when they have 5 pounds of wool on them, how many days will it take for one Gnorbu to supply 100 pounds of wool? Please round up to the nearest full day.

Possible Methods:

month of Celebrating: Dec (31 days)

month of Sleeping: Jan (30 days)

month of Awakening: Feb (28 or 29 days)

month of Running: Mar (31 days)

month of Eating: Apr (30 days)

month of Hunting: May (31 days)

month of Relaxing: Jun (30 days)

month of Swimming: Jul (31 days)

month of Hiding: Aug (31 days)

month of Gathering: Sep (30 days)

month of Collecting: Oct (31 days)

month of Storing: Nov (30 days)

Number of years needed = 100/((2+1.2+0.6+1.5)*3) = 6.289

Amount of wool left after 6 years = 100 - 6*(2+1.2+0.6+1.5)*3 = 4.6

Number of months required to get the remaining wool = 2

Number of days required to get the remaining wool = 0.6/(1.2/31) = 16

Total number of days required = 4*365 + 2*366 + 31 + 28 + 16 = 2267

Answer: 2267

 Lenny Conundrum Round 378

Kauvara is making a special potion. For this particular potion, she's filling a bottle in the shape of a pentagonal pyramid: the base is a regular pentagon, 15cm to a side, and the height of the "pyramid" is 32cm. Assume these measurements are for the interior of the bottle.

Assume the flow of potion into the bottle starts at zero and increases at a rate of 1 millilitre per second per second. How many seconds will it take to completely fill the bottle? Please round to the nearest whole number.

Possible Methods:

Volume of pyramid = 1/3*Base Area*Height

= 1/3*(Side Length)²*Number of Sides/(4*tan(pi/Number of Sides))*Height

= 1/3*15²*5/(4*tan(pi/5))*32

= 4129.145761 cm³

dV²/dt² = 1

dV/dt = f 1 dt = t

V = f t dt = t²/2

t = f 2V = f (2*4129.145761) = 90.88 s

Answer: 91

Lenny Conundrum Round 375 

Suppose, at a certain festival, they are serving pie. The pies are flat on the bottom and flat on the top. The top is a circle with a radius of 5.5 inches. The bottom is a circle with a diameter of 8.5 inches. The height of the pie is 2.8 inches. Otherwise, they're all completely normally pie shaped.

There are 375 pies, and, for some strange reason, they're all made out of solid granite. If granite weighs exactly 171.6 pounds per cubic foot, how much do all 375 of the pies weigh, put together? Please round to the nearest whole number, and just submit a number as your answer, with no other information.

Possible Methods:

The pie here is basically an inverted Conical Frustum and 1 inch = 0.0833 foot

Volume of inverted Conical Frustum = 1/3 * pi * height * (bottom radius² + top radius² + bottom radius * top radius)

= 1/3 * pi * (2.8 * 0.0833333333) * ((8.5/2* 0.0833333333)² + (5.5* 0.0833333333)² + 8.5/2* 0.0833333333*5.5* 0.0833333333)

= 0.1216427806 cubic foot

Total weight of pies = 375 * 0.1216427806 * 171.6 = 7828 pounds

Answer: 7828