NeoPets Riddles (Lenny Conundrums) and Answers Here

Um, hi everybody
I'm new to this site, sooo i dont know how to post a thing without it being a reply to somebody, soo if this is like an inconvenience in anyway, im sorry
What formulas would you use to figure this out?
I'm not really interested in the prizes, just the challenge
And, i havent learned how to do this yet in school, im only a freshman in hs, but im in geometry.
I read that you guys used the law of sines and such but my teacher hasnt taught us how that applies to cirlces yet, just triangles. Soo, does anybody have any ideas where i could begin? Could i make a scale? Like, the circle has a diameter of 2000inches but a measured diameter of 1 and 11/16inches. I think im off a little on that measurement, but could i do that? Another question, is it an exact answer or are we rounding? because if you use the formula for the area of a circle, its pi*(radius)squared. Soo... youd get a decimal right? Rawr, sorry for asking so much, thank you to anybody that helps

Hi crazyrock,

1. At the top and bottom of the page, there are Reply to All tabs that you can use to post without replying to anyone in particular.

2. We are applying the law of sines and cosines to triangles. (And I'm surprised that you've seen them at all, since they depend on trigonometric functions.) Anyway, the idea is to draw triangles between the centers of the different circles. First, I drew one between the centers of the large circle and two adjacent medium size circles. Then I can express the lengths of the three sides in terms of the radius R of the medium size circles and the known (1000) radius of the large circle. Based on the number of medium sized circles around the large circle, I can determine the size of the angles in the triangle. Then I can use the law of sines to get an equation and solve for R.

Once I have R, then I create another triangle between the centers of three adjacent circles of three different sizes. I can express the lengths of the sides in terms of the three radii and I can get the angle at the point at the center of the large circle (based on the geometry again). Then I use the law of cosines to get an equation and solve for r (the radius of the small circle).

3. For the second step, you can also search for Soddy Circles. Each small circle is a Soddy Circle in regards to the large circle and two medium circles. There is a formula to calculate the radius of the small circle in terms of the radii of the other three circles.

4. The final answer is definitely a decimal. I rounded it to the nearest whole square inch to submit it. Normally, they tell you to do that in the LC, so I'm guessing that's what they want. Otherwise, who knows what the appropriate number of decimals to submit is? I got different values after the decimal point depending on whether I used Excel or my calculator.

You can PM (private message) by clicking on a user's name and using the Send Message link on the right side.

Yeah. don't be like me and accidentally reply instead of PM a person

Hey guys... long time no see!

Thank you so much
I have no idea why sine or any other trigonometric functions work, my geometry teacher has only introduced them and we've used them a few times but nothing as involved as this.
I tried it, to my best ability.. Ha, I never thought I'd be doing math over spring break, haha, first night home and this is what I do Oh well, I tried my hardest. And I really hope its correct, though, I'm assuming it's probably wrong, I wouldn't be surprised Well, if it's alright with you, I'll private message you my answer.. I'm sure it wont be anything near what you got, but it's worth a try
I never thought there would be such hard problems on neopets!
Anyways, thank you so much, I really do appreciate it

Oh jeeze, nevermind. I know I definitely screwed it up.. Grr, I just spent like at least 45 minutes on that
Oh well, my radius came out bigger than the radius of the big circle sooo, yeah, thats more than I can do at my level. Thank you so much for taking the time to help me, I really appreciate it, like a lot

i made a radom stupid guesss.

Oh, yeah, they have some really involved problems in the Lenny Conundrum. I'd say the minimum level to be able to handle all the math problems is probably three years of high school math. And then they apply the concepts in novel ways. This week it was Soddy Circles. We've also seen spherical caps, exponential growth, and all sorts of other things. And then there are the ones where you have to find a pattern somewhere on the Neopets site. Those don't require math knowledge, but I find them even tougher in a lot of cases.

Starting to look like they are going to go for another week. The conundrum appears to still be open. Anybody gotten mail?

no, nothing yet. nothing on the news page either

Well, it is past 2pm NST. Definitely looks like there won't be a new LC this week. I expect the news will confirm it when it comes out. See everyone next week.

"The Lenny Conundrum competition is continuing on for another week. "

from the news page.

I saw that in the news too. I wonder why they did that. Do you think that not many people were able to solve the problem?
Have they ever held over a competition before?

yes, there have been many extensions before. not sure what the reason might be, but i don't think it's the number of correct replies. would be nice, though...

Congratulations! You have guessed correctly in the Lenny Conundrum game (round 352). You have won 1258 NP!

Because you were in the first 250 to guess correctly, you also have been awarded a Pink Flower Bouquet, and receive a trophy and the Lenny Conundrum avatar!

Yours Sincerely,
The Neopets Team!

Solution to last week's puzzle:

We know the radius of the large circle = 1000 in. I'm going to call the radius of the medium circle R and the small circle r.

To calculate R, I used the law of sines. This says that for any circle, the ratio of the length of a side over the sine of the opposite angle is equal to the same ratio for any other angle. If a, b, c are the lengths of the sides and A, B, C are the opposite angles, then:

a / sin A = b / sin B = c / sin C

I took a triangle joining the center of the large circle with the centers of two adjacent medium circles. The sides of this triangle will pass through the tangent points of the circles. Call the center of the large circle C and the centers of the medium circles c1 and c2. Then sides C-c1 and C-c2 are 1000 + R inches long. Side c1-c2 will be 2R inches long. Since the 10 medium circles are spaced evenly around the large circle, the angle C will be 36 (360 / 10) degrees and since we have an isoceles triangle, the other two angles will each be 72 degrees (since the sum of the interior angles of a triangle is 180 degrees).

Then we have (1000 + R) / sin 72 = 2R / sin 36. Solving for R, we get R = 1000 sin 36 / (2 sin 72 - sin 36). Evaluating this gives us R = 447.2135955 (in Excel, anyway).

To find r, we need to use the law of cosines. Using a, b, c as the lengths of the sides and A, B, C as the opposite angles, this says:

c^2 = a^2 + b^2 - 2ab * cos C

(When C is 90 degrees, cos C becomes 0 and this simplifies to the Pythagorean Theorem for right triangles.)

Now I created a triangle from the center of the large circle to the center of one of the medium circles and then to the center of one of the adjacent small circles. This gives me a triangle with sides of 1000 + R, 1000 + r and R + r. The angle at the center of the large circle (make this C in the law of cosines) will now be 18 (360 / 20) degrees.

So, (R + r)^2 = (1000 + R)^2 + (1000 + r)^2 - 2(1000 + R)(1000 + r) * cos 18. We then multiply everything out to get:

R^2 + 2Rr + r^2 = 10^6 + 2000R + R^2 + 10^6 + 2000r + r^2 - 2*10^6*cos(18) - 2000R*cos(18) - 2000r*cos(18) - 2Rr*cos(18).

Canceling R^2 + r^2 from both sides and moving all the terms with r to the left, we get:

2Rr - 2000r + 2000r*cos(18) + 2Rr*cos(18) = 2*10^6 + 2000R - 2*10^6*cos(18) - 2000R*cos(18).

Solving for r, we get:

r = (2*10^6 + 2000R - 2*10^6*cos(18) - 2000R*cos(18)) / (2R - 2000 + 2000*cos(18) + 2R*cos(18))

and Excel gives me r = 86.0029877

So the area of one small circle is PI * r^2 = 23236.83371 and we have ten of them to give us the final answer of 232368.3371. I rounded my submission to 232368 as they commonly ask for the nearest integer value in the LC, although they didn't specify that this week.

Ugh. I entered 232368, not right though apparently. I must have made a typo.

it's outttttttttttttttt and i have no clue


The current prize for the first 250 correct entries is a Potion of Brightvale Castle. You will also get a cool trophy that people can see when they do a user lookup on you! And all correct entries will split the 2,000,000 Neopoint prize pool.

THIS WEEKS PUZZLE - ends Next Wednesday, most likely

What is the missing word or phrase in this sequence?

Snowflake
Tree Pattern
Star Shaped
Mistletoe
__________
Bubble
Rainbow

Please submit only the answer with no other information.

seems that all are stained glass windows