## Wednesday, July 18, 2012

### GeekDad Puzzle of the Week: Math Trolls

GeekDad Puzzle of the Week: Math Trolls:
Q: Nora is taking a trip to visit her Grandmother in northernmost New York State this week, to bring her some freshly picked berries. On the way there, she has to cross a total of 30 bridges, and under each of the these bridges lives a troll. Each troll is aware of their bridge number, and either demands or gives berries based upon the rarest or most applicable description of their bridge. They demand or give berries according to the following schedule:
• Trolls under odd numbered bridges demand half of your berries.
• Trolls under even numbered bridges demand 20 berries.
• Trolls under prime numbered bridges give you half again the number of berries you are carrying.
• Trolls under perfect square numbered bridges demand a quarter of your berries.
• Trolls under perfect cube numbered bridges give you the number of berries you are carrying, doubling your number of berries.
If trolls round up in their demands (i.e., if you have 57 berries at the foot of a bridge best described as odd numbered, you will cross it with 28 berries), what is the minimum number of berries Nora must start with so that she ends up with 1,000 berries when she arrives at her Grandmother’s house?
The solution has already been posted on the GeekDad website. If you want to solve it yourself, read no further. A detailed breakdown of my solution is given in the comments.

For this puzzle, I'll assume the demands are ordered from most common to rarest. In other words, if a bridge is odd, a perfect square and a perfect cube, the last one is the most applicable.

The easiest way to solve this is to just work backwards from 1000 berries. At each bridge you perform the reverse operation until you get back to bridge 1. While the puzzle says trolls will round demands in their favor, it doesn't say what they will do when giving you half again the number of berries you are carrying. I worked it both ways (rounding down and rounding up) and still got the same answer of 44 berries. For the example below, I'll assume trolls always round to their advantage (round up when demanding berries, round down when giving berries).

1) Cube - double the berries 44 x 2 = 88
2) Prime - add half your berries 88 + 44 = 132
3) Prime - add half your berries 132 + 66 = 198
4) Square - take a quarter 198 - 50 = 148
5) Prime - add half your berries 148 + 74 = 222
6) Even - take 20 berries 222 - 20 = 202
8) Cube - double the berries 303 x 2 = 606
9) Square - take a quarter 606 - 152 = 454
10) Even - take 20 berries 454 - 20 = 434
11) Prime - add half your berries 434 + 217 = 651
12) Even - take 20 berries 651 - 20 = 631
13) Prime - add half your berries 631 + 315 = 946
14) Even - take 20 berries 946 - 20 = 926
15) Odd - take half your berries 926 - 463 = 463
16) Square - take a quarter 463 - 116 = 347
17) Prime - add half your berries 347 + 173 = 520
18) Even - take 20 berries 520 - 20 = 500
19) Prime - add half your berries 500 + 250 = 750
20) Even - take 20 berries 750 - 20 = 730
21) Odd - take half your berries 730 - 365 = 365
22) Even - take 20 berries 365 - 20 = 345
23) Prime - add half your berries 345 + 172 = 517
24) Even - take 20 berries 517 - 20 = 497
25) Square - take a quarter 497 - 125 = 372
26) Even - take 20 berries 372 - 20 = 352
27) Cube - double the berries 352 + 352 = 704
28) Even - take 20 berries 704 - 20 = 684
29) Prime - add half your berries = 684 + 342 = 1026
30) Even - take 20 berries 1026 - 20 = 1006

2. Do you know about our Aurora Bridge Troll here in Seattle?
Google Fremont Troll or Aurora Bridge Troll or Seattle Troll for photos. I can't upload them here.

1. In fact, when I was looking for pictures of "bridge troll" quite a few of them were of the Aurora Bridge Troll.

2. Did you happen to notice that the troll has a real VW Beetle in it's clutches? Harder to notice is one eye is a VW wheel cover. This troll is located beneath the North end of the Aurora Bridge on Hy 99 that runs from Canada to Mexico, or maybe it's the other direction, and if you were to ask almost anyone here in Seattle where the George Washington Bridge is, you would get blank looks, but that is the true name of this major bridge we all call Aurora. Also, a short distance to the West of the troll is a huge bronze statue of Lenin that someone managed to get out of Russia after the collapse of the Soviet Union. Quite ironic to see an enormous statue of Vladimir Lenin in this country.

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You may provide indirect hints to the answer to show you know it, but make sure they don't assist with solving. You can openly discuss your hints and the answer after the deadline. Thank you.