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Icosahedron Diamond Found!!

Posted by Karl Sharman on May 20, 2011 – 12:05 am

The wily explorer Knightmare, and his trusty companion, Slavy have scoured the deserts of the world in search of the fabled Icosahedron Diamond. Hundreds of miles from civilization, they discover the aforementioned bauble, a beautiful giant crystal worth a King’s ransom.

Unfortunately, the diamond is on a tiny island in the centre of a perfectly square lake. Fortunately, Knightmare and Slavy have been taking swimming lessons….. unfortunately, the lake is not water but highly concentrated acid, the result of ignoring paranoid environmentalists.
The lake’s sides are 40 feet long, and shortest distance from the middle of a side to the island is a little over 19 feet. Conveniently close to the lake there are a half-dozen wooden planks, each measuring 16 feet by 16 inches by 2 inches. They’re just few feet too short to bridge the gap – Curses! If only they’d brought some tools along on this expedition- rope, nails, super glue, anything!
Swearing to actually plan their next expedition they sit in the sand and try to puzzle their way out of this quandary.
Will they walk back to civilization empty-handed, or can you help them think of a way to the Icosahedron? In the traditional manner of puzzles, they can only use the planks of wood, they have no tools, and modesty prevents the use of clothing etc!!!
Bonus points for how many faces, points and edges on an Icosahedron….. not that it helps get the diamond home, but it’s a scant comfort if you can’t get the diamond!

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10 Responds so far- Add one»

  1. 1. Bob Said:

    Bridge over each corner at 45 angle, then bridge from middle of each of these planks to the bank again, then bridge between the middle of these two planks and the last plank should reach the diamond.

  2. 2. John24 Said:

    Starting on the side of the pond closest to the island, lay one plank A at a 30 degree angle accross one corner of the pond forming a 60 degree angle on the adjacent side. Place another plank B on the corner adjacent to the 60 degree angle formed by the first plank. This creates 2 60 degree angled planks next to each other.

    Lay one plank C from plank A to the side of the pond between plank A and plank B at exactly 10 ft on plank A and 10 ft from plank A on the side of the pond.
    Lay one plank D from plank B to the side of the pond between plank A and plank b at exactly 10 ft on plank B and 10 ft from plank B on the side of the pond.
    Each of these 2 planks C & D form 2 20, 60, 90 triangles which you can use to determine the distance from shore for the next plank.

    Lay a plank E from plank C to plank D at a point 3+ feet from shore.

    Finally, lay plank F from plank E to the island.

    I actually believe you could reach the island with 3 planks but not sure the distance from the island to the corner of the pond or the exact shape of the island.

    icosahedron 20 sides with 12 vertices.

  3. 3. John24 Said:

    30 edges

  4. 4. Knightmare Said:

    hooray! i’m rich!

    now Slavy and i have a long trip home. one treasure for two men…sure hope there won’t be any “accidents”.

    btw: don’t need to swim – i float.

  5. 5. Karl Sharman Said:

    Obviously way too easy… Bob gets the diamond, and for a small bit of solace John24 knows what an Icosahedron looks like, and can amuse his friends at his next dinner party with an in-depth description.
    Knightmare – you need to watch out – silent assassins are the worst, and Slavy’s awfully quiet at the moment…..

  6. 6. John24 Said:

    The only reason I know what an Icosahedron looks like is due to a parent on my son’s soccer team. This parent showed us how a soccer ball’s unique color pattern is actually an icosahedron with the vertices cutoff forming the black pentagons. Truncated icosahedron is the term given to this unique shape.

    12 = # of black pentagons on a soccer ball (at each of the missing vertices)

    20 = original # of sides
    32 = modified # of sides

    30 = original # of edges
    90 = modified # of edges

  7. 7. JG Said:

    I’VE GOT THE ANSWER!!! Its a tundra. Tundras are deserts. The acid is frozen. Walk over the frozen acid.

  8. 8. oge Said:

    use slavy as an extra plank

  9. 9. HamsterofDeath Said:

    it’s in a desert. just keep throwing sand into the lake until you created a bridge out of sand. who needs the planks?

  10. 10. cazayoux Said:

    Hello!

    This problem snuck back into my noodle at the end of last week.

    It doesn’t seem that a full solution was delivered.

    What is the furthest into the pond that you can reach with 1 board? with 2 baords? 3? 4? 5? n?

    Maybe my approach was too complicated, but I didn’t see an easy way to do this.

    One board is straightforward.
    If the corner is seen as a cartesian grid (0,0), then the 16 foot board can take every possible position from [(0,16),(0,0)] to [(0,0),(16,0)].
    At 11.31 feet [sqrt(16*16/2)], we are 8 feet from the origin.

    Two boards isn’t difficult yet.
    For every position (with sufficient granularity) of board 1, I can consider every possible position of board 2.

    Three boards starts getting a little more complicated.
    I have to consider board three spanning the y-side land and board 1, the y-side land and board 2, board 1 and board 2, board 1 and the x-side land, and board 2 and the x-side land.

    With 5 boards, how far CAN we get from the corner?

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