This is a puzzle that has to be solved to find a geocache, and I'm stumped.



Flower Power

Geocache Description:

The cache is obviously not at the posted coordinates, but is within 1-2 miles of the pond. This is an easy puzzle if you think it through, and is based purely on math. It is really long though (so long there wasn't a FTF until May), and I dare you to do this without a calculator. Since the dandelions are popping up around this time, I figured it would be right to make a puzzle on dandelions, and since I came up with this problem and solved it about a year ago, I decided to adapt it to cache use.

Suppose there is one dandelion in the middle of a field. That dandelion will produce 200 seeds for that year. Suppose all of the seeds ever given off are certain to grow, all the offspring give off exactly 200 seeds the next years, and none of the dandelions ever die. How many dandelions will there be in field on the 100th year?

Once you have this number, take the 26 leftmost digits on the number. In order from left to right, they are A-Z. The coordinates for the cache are N41 CT.TMS W070 JG.HVQ. Good luck, and have fun!

Any help is appreciated!

Hugs, Devlyn

Each year, you have 201 times as many plants as the year before: the 200 new ones for each original one, plus the original ones that are still there.  So, in year n, you have 201^(n-1) plants.

In year 100, you have 201^99 plants.  That's a really big number.  I can't even count the digits because, on my screen, it is wider than the full screen, but it's around 250 digits long.  You need to calculate at least the first 26 digits to get the full alphabetic translation required for the coordinates.  You don't actually use all 26 letters, but you have to get as far as the letter 'T', which means 20 digits.  I tried the calculation in Excel, but it pooped out after 15 digits.

So, no, I can't get you the coordinates.

No calculator, my ***!

http://rabbit.eng.miami.edu/class/een218/calculator.html

I'd replace 200 with 3 and figure out the equation that way so you can validate. Then put the 200 back in there. Don't forget the one you started with.

only the offspring give seed ... but the others remain ...

1
200      +1            200^1 + 200^0
200 * 200   +200 +1            200^2 + 200^1 + 200^0
200*200*200   +200*200 +200 +1      200^3 + 200^2 + 200^1 + 200^0

There is a formula for that ...


hugs

The 200 is a trick, just use 2 since you ignore all but the first 26 digits. They only reproduce once. The rest looks straightforward.

Quote from: AnonyMs on May 04, 2016, 10:50:21 AMThey only reproduce once.

Hmm.  I see that it could be interpreted that way.  I hate ambiguous puzzles!

Quote from: KathyLauren on May 04, 2016, 11:05:43 AM
Hmm.  I see that it could be interpreted that way.  I hate ambiguous puzzles!

It's probably meant to be easy, once you understand it.

Quote from: KathyLauren on May 04, 2016, 10:08:46 AM
Each year, you have 201 times as many plants as the year before: the 200 new ones for each original one, plus the original ones that are still there.  So, in year n, you have 201^(n-1) plants.

In year 100, you have 201^99 plants.  That's a really big number.  I can't even count the digits because, on my screen, it is wider than the full screen, but it's around 250 digits long.  You need to calculate at least the first 26 digits to get the full alphabetic translation required for the coordinates.  You don't actually use all 26 letters, but you have to get as far as the letter 'T', which means 20 digits.  I tried the calculation in Excel, but it pooped out after 15 digits.

So, no, I can't get you the coordinates.

No calculator, my ***!

This looks really good and gives coordinates right on the edge of a bike path in the target area. Will let you know when I get verification or find my way down there.

Thanks for all the help, everyone! 

Hugs, Devlyn

I'm not sure the 201 is accurate. Wouldn't it be the algorithm using 200, then adding a +1 after the calculation? Each of the 200 will reproduce, but the original one has already done it.

I also just tried it with 200 and 2. With scientific notation, the displayed digits are identical between the two with the only difference being the total number of digits, 228 and 30 respectively. Using 2 has enough digits to get the full 26. Whoever said that... genius!

Quote from: KathyLauren on May 04, 2016, 10:08:46 AM
Each year, you have 201 times as many plants as the year before: the 200 new ones for each original one, plus the original ones that are still there.  So, in year n, you have 201^(n-1) plants.

In this formula its 201 squared.
But only 200 are reproduced, the old ones do not reproduce.

It needs to be a sum imo.

The old ones plus the new ones.

Imo its a geometrical line, and there is a formula for that.

Unless there is some kind of hidden formulation which makes it all a diiferent thing.

"all the offspring give off exactly 200 seeds the next years"
meaning there is 200 offspring always every year ?


hugs

Quote from: Emileeeee on May 05, 2016, 08:30:02 AM
I'm not sure the 201 is accurate. Wouldn't it be the algorithm using 200, then adding a +1 after the calculation? Each of the 200 will reproduce, but the original one has already done it.

As noted above, it depends how you interpret the question.  It doesn't actually state that the plants only reproduce once.  I know that my dandelions reproduce every year!!

If the plants only reproduce once each, then the answer is 200^99 +1 (or 2^99 +1 since you only need the first 26 digits).  If the plants reproduce every year, then it is 201^99.  The bit about doing it without a calculator might be a hint about which way to interpret it, since 2^99 is feasible (though tedious) without a calculator.  201^99 is not.

Quote from: KathyLauren on May 05, 2016, 08:58:29 AM
It doesn't actually state that the plants only reproduce once. 

"That dandelion will produce 200 seeds for that year."

Yes. meaning could be : only for that year or each year.
With only for that year its a geometrical line.
If the formula is used then results are over 60 degrees ,and even if recacculated for degrees the result is in the sea.

So each year might be a possibility.
But without calculator ?


hugs

Quote from: Laura_7 on May 05, 2016, 09:07:11 AM
But without calculator ?

For 2^99, sure.  Computer nerds know that 2^16 = 65536.  So 2^99 = 8 * 65536^6.  That would be a pain in the butt to do longhand, but only involves 6 multiplications.

Quote from: KathyLauren on May 05, 2016, 08:58:29 AM
As noted above, it depends how you interpret the question.  It doesn't actually state that the plants only reproduce once.  I know that my dandelions reproduce every year!!

If the plants only reproduce once each, then the answer is 200^99 +1 (or 2^99 +1 since you only need the first 26 digits).  If the plants reproduce every year, then it is 201^99.  The bit about doing it without a calculator might be a hint about which way to interpret it, since 2^99 is feasible (though tedious) without a calculator.  201^99 is not.

Ah yes it is ambiguous. I hadn't read it that way, but I do see that it could mean that each one produces 200 offspring each year. What a nightmare. I'd skip that one

I have the one set of promising coordinates, do any other math whizzes want to give this a shot before I commit to a hundred mile ride? 

Hugs, Devlyn

I think you should go with two sets of coordinates, one derived from 2^99, and the other from 201^99.  The two sets of 26 digits are:
63382530011411470074835160
10385076131385329514228303

Only one set is on land, though! 

Hugs, Devlyn

Quote from: Devlyn Marie on May 07, 2016, 09:30:24 AM
Only one set is on land, though! 

Hugs, Devlyn

LOL!  I guess that would make a difference! 

Quote from: Devlyn Marie on May 07, 2016, 08:31:14 AM
I have the one set of promising coordinates, do any other math whizzes want to give this a shot before I commit to a hundred mile ride? 

Hugs, Devlyn

I agree with the solutions posted so far, and my calculator agrees with Kathy's.

In [1]: print `201**99`[:26]
Out[1]: 10385076131385329514228303

Enjoy your ride!