Lenny Conundrum Round 373
For the Chocolate Ball, a magnificent new sweet was created. It was made by taking a 4-centimetre-diameter sphere of nougat and coating it with 1-centimetre-thick layer of marzipan. Then the whole thing is covered in a layer of caramel, which is then coated with a 2-centimetre-thick layer of dark chocolate.
Nougat has exactly 21.3 calories per cubic centimetre, and marzipan has 24.1 calories per cubic centimetre, and caramel has 18.7 calories per cubic centimetre, and dark chocolate has 16.2 calories per cubic centimetre. If the caramel makes up exactly 40% of the total calories of the sweet, how many total calories is this particular sweet? Please round to the nearest whole number, and submit only a number with no other information.
Possible Methods:
Volume of nougat = 4/3*pi*(4/2)3 =32?/3
Volume of marzipan = 4/3*pi*(6/2)3 - 4/3*pi*(4/2)3 =76pi/3
Volume of caramel = 4/3*pi*(x/2)3 - 4/3*pi*(6/2)3 =(0.5x3- 108)pi/3
Volume of dark chocolate = 4/3*pi*((x+4)/2)3 - 4/3*pi*(x/2)3 =(6x2+24x+32)pi/3
(0.5x3- 108)pi/3 *18.7 = 40/100*(32pi/3*21.3+76pi/3*24.1+(0.5x3- 108)pi/3*18.7+(6x2+24x+32)pi/3*16.2)
9.35x3-2019.6 = 272.64 + 732.64 + 3.74x3-807.84+38.88x2+155.52x+207.36
5.61x3-38.88x2-155.52x-2424.4 = 0
Using Cubic Equation Calculator, x= 12.143753212769147.
Therefore, total calories of the sweet = 100/40*(0.5*12.1437532127691473- 108)pi/3 *18.7 = 38549.52
Answer: 38550
Lenny Conundrum Round 369
A Moehog entered a certain place of business. After his transaction was completed, he was handed a receipt, which simply had the number "664" circled on it.
Later, a Koi entered the same place, and left with a recipt, with the number "644" on it.
An Ogrin entered next, and her receipt had the number "651".
A Mynci, a Flotsam, and an Acara then completed their transactions at the establishment, and each received a receipt.
If you totaled the numbers on all six receipts, what would the sum of all six receipts be? Please submit only a number, with no punctuation, words, or any other information other than your answer. For example, if the three unknown receipts all had the number 100, your submitted answer should be "2259".
Possible Methods:
1. Sum of 3 digits follows an even number pattern and is always arranged from biggest number to smallest number. In addition, the first digit remains 6 and the last digit must be the same for 2 consecutive numbers.
664: 6+6+4=16
644: 6+4+4=14
651: 6+5+1=12
631: 6+3+1=10
620: 6+2+0=8
600: 6+0+0=6
Sum of the 6 numbers = 664 + 644 + 651 + 631 + 620 + 600 = 3810
2. Sum of last 2 digits follows an even number pattern and is always arranged from biggest number to smallest number. In addition, the last digit must be the same for 2 consecutive numbers.
664: 6+4=10
644: 4+4=8
651: 5+1=6
631: 3+1=4
620: 2+0=2
600: 0+0=0
Sum of the 6 numbers = 664 + 644 + 651 + 631 + 620 + 600 = 3810
Answer: 3943
Lenny Conundrum Round 367
In the Royal Neopian Neolodge, your nightly hotel price gets you a bedroom and a bathroom. Each room is perfectly square. If you take the length of one of the walls of the bedroom, and subtract the length of one of the walls of the bathroom, you'll find that the difference is 9 metres. Now, if you multiply the length of the bedroom wall by the length of the bathroom wall, you'll find that the product is 12 square metres.
What is the total area, in square metres, of the bedroom plus the bathroom? Round to the nearest whole number, if necessary, and please submit only a number.
Possible Methods:
Since both rooms are perfectly square, all sides of the rooms are equal.
Let x be the side of bedroom and y be the side of bathroom.
x-y=9
x=9+y (1)
xy=12 (2)
Sub (1) into (2)
(9+y)y=12
y2+9y-12=0
y=(-9+?(92-4(1)(-12)))/2
y=(-9+?129)/2
x=9+(-9+?129)/2
x2+y2=(9+(-9+?129)/2)2+((-9+?129)/2)2
x2+y2=105
Answer: 105
Lenny Conundrum Round 343
Suppose you have a scale model of Neopia. The model is a perfect sphere. The diameter of the sphere is exactly 41 inches, and is constructed of solid steel, exactly 0.8 inches thick. Inside the sphere, there are 50 smaller steel spheres, each exactly 2 inches in diameter, and the rest of the sphere is filled up with water.
If the density of steel is exactly 490 pounds per cubic foot, and the density of water is exactly 62.4 pounds per cubic foot, how much does the model weigh, in pounds? Please round to the nearest whole pound, and submit only a number with no other information as your answer. For example, if you get the answer 314.159 lbs, you would submit the answer "314".
Possible Methods:
Method 1
Radius of large sphere = 41/2*0.0833333333=1.708333327 foot
Thickness of steel = 0.8*0.0833333333=0.066666666 foot
Volume of large sphere = 4/3*pi*(1.708333327)^3 =20.88365207 cubic foot
Volume of large steel sphere = 20.88365207-4/3*pi*(1.708333327-0.066666666)^3=2.350745124 cubic foot
Radius of small sphere = 2/2*0.0833333333=0.0833333333 foot
Volume of a small steel sphere =4/3*pi*(0.0833333333)^3-4/3*pi*(0.0833333333-0.066666666)^3=0.002404675 cubic foot
Mass of steel spheres = (2.350745124+0.002404675*50)*490 = 1210.779669 lbs
Mass of water = (20.88365207 - 2.350745124 - 0.002404675*50)*62.4= 1148.950807 lbs
Total mass = 1210.779669 lbs + 1148.950807 lbs = 2360 lbs
Method 2
Radius of large sphere = 41/2*0.0833333333=1.708333327 foot
Thickness of steel = 0.8*0.0833333333=0.066666666 foot
Volume of large sphere = 4/3*pi*(1.708333327)^3 =20.88365207 cubic foot
Volume of large steel sphere = 20.88365207-4/3*pi*(1.708333327-0.066666666)^3=2.350745124 cubic foot
Radius of small sphere = 2/2*0.0833333333=0.0833333333 foot
Volume of a small steel sphere =4/3*pi*(0.0833333333)^3=0.002404068 cubic foot
Mass of steel spheres = (2.350745124+0.002404068*50)*490 = 1211.254786 lbs
Mass of water = (20.88365207 - 2.350745124 - 0.002404068*50)*62.4= 1148.8903 lbs
Total mass = 1211.254786 lbs + 1148.8903 lbs = 2360 lbs
Answer: 2360
Lenny Conundrum Round 20
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?
Possible Methods:
Let amount of NP red, green purple and silver coins represent be R, G, P, and S respectively.
Since three red coins plus two Neopoints equal a green coin,
3R + 2 = G (1)
Since three green coins plus five Neopoints equal a purple coin,
3G + 5 = P (2)
Since eleven red coins plus a single Neopoint equal a purple coin,
11R + 1 = P (3)
Since silver coin is worth four purple coins,
S = 4P (4)
Substitute (1) into (2), 3 x (3R + 2) + 5 = P => P = 9R + 11 (4)
(3) - (4), 2R - 10 = 0 => R = 5
Substitute R=5 into (3), P = 11 x 5 + 1 = 56
Substitute P=56 into (4), S = 4 x 56 = 224
Total NP change = 224 - 84 = 140 NP
Total number of red coins change = 140 / 5 = 28
Answer: 28 red coins (red=5np, green=17np, purple=56np, silver=224np)
Lenny Conundrum Round 18
Heermeedjet the Meerca has 5,000 Neopoints in his pocket (after successfully breaking into somebody's Neohome). He has a go on the Wheel of Excitement and then increases the amount of money he has by 20%. He then takes 20% of his money and spends it on Neggs.
Heermeedjet then puts the rest of the money in the National Neopian Bank, and earns 1% interest every day (daily interest is rounded down) for 10 days. After 10 days he gets bored and puts the all the money on a PoLogle at 3:1 odds and wins.
How much money does Heermeedjet have left?
Possible Methods:
Total Amount Left after Wheel of Excitement = 5000 / 100 x (100 + 20) = 6000 NP
Total Amount Left after Spending on Neggs = 6000 / 100 x (100 - 20) = 4800 NP
Day 1: Total Amount Left after Bank Interest = 4800 / 100 x 101 = 4848 NP
Day 2: Total Amount Left after Bank Interest = 4848 / 100 x 101 = 4896 NP (Round Down to Nearest Whole Number)
Day 3: Total Amount Left after Bank Interest = 4896 / 100 x 101 = 4944 NP (Round Down to Nearest Whole Number)
Day 4: Total Amount Left after Bank Interest = 4944 / 100 x 101 = 4993 NP (Round Down to Nearest Whole Number)
Day 5: Total Amount Left after Bank Interest = 4993 / 100 x 101 = 5042 NP (Round Down to Nearest Whole Number)
Day 6: Total Amount Left after Bank Interest = 5042 / 100 x 101 = 5092 NP (Round Down to Nearest Whole Number)
Day 7: Total Amount Left after Bank Interest = 5092 / 100 x 101 = 5142 NP (Round Down to Nearest Whole Number)
Day 8: Total Amount Left after Bank Interest = 5142 / 100 x 101 = 5193 NP (Round Down to Nearest Whole Number)
Day 9: Total Amount Left after Bank Interest = 5193 / 100 x 101 = 5244 NP (Round Down to Nearest Whole Number)
Day 10: Total Amount Left after Bank Interest = 5244 / 100 x 101 = 5296 NP (Round Down to Nearest Whole Number)
Total Amount Left after Winning = 5296 x (3 + 1) = 21184 NP
Answer: 21,184 Neopoints
Lenny Conundrum Round 16
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???
Possible Methods:
Assuming there are only 3 types of Grundo (Green, Purple and Red) and let number of Grundos be G, P and R respectively,
Since all but two of the Grundos were Green,
P + R = 2 (1)
Since all but two of the Grundos were Purple,
G + R = 2 (2)
Since all but two of the Grundos were Red,
G + P = 2 (3)
(1) - (2), P - G = 0 => P = G
Substitute P = G into (3), P = G = 1
Substitute P = 1 into (1), R = 1
Total number of Grundos = G + P + R = 1 + 1 + 1 = 3
Answer: 3