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Guest Message by DevFuse
 

Rainman

Member Since 30 Jul 2012
Offline Last Active Sep 12 2014 01:57 AM
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Topics I've Started

Inequalities for large integers

29 January 2014 - 12:49 PM

These problems use Knuth's up-arrow notation (http://en.wikipedia....-arrow_notation), with ^ being the up-arrow.

 

1. Find the smallest value for n, such that 1000^^n > 12^^(n+1).

 

2. We define the sequence g, by assigning g(1) = 3^^^^3, and g(n+1) = 3^^^...^3, where the number of up-arrows is g(n). A famously large number is Graham's number = g(64). Which is larger, Graham's number or 2^^^...^2, where the number of up-arrows is Graham's number?


Chase on an endless road, continued

30 July 2013 - 01:36 PM

This is a sequel to the puzzle: http://brainden.com/...n-endless-road/

 

You have just managed to catch up to your friend in a race, on a road which goes on forever in both directions. Your friend is a bad loser, and full of adrenaline from the race. "Try and catch me this time", your friend says, and knocks you out cold. When you recover, you have no idea how much time has passed.

  • You move at constant speed x>1 m/s and your friend moves at constant speed 1 m/s.
  • Visibility is limited so you will not be able to see your friend from a distance.
  • You must not stray from the road, same goes for your friend.

Can you catch up to your friend? Does the answer depend on your speed x, and if so, what is the minimum x for which you can catch up? What is your strategy?


Fourteen sacks of gold

30 July 2013 - 01:01 PM

A treasure hunter has just found 14 sacks of gold on a pedestal in a temple. The problem is, by taking the gold sacks he awakened the two guardians of the temple and was captured. The guardians have agreed to spare his life if he can give to both of them the same non-zero amount of gold, without opening the sacks.

  • Each sack is marked with a distinct 3 digit number (anything from 001 to 999) and contains that much gold.
  • After the guardians have received their gold, the treasure hunter must place all remaining gold sacks back on the pedestal.

To make this more difficult for you, I will not tell you how much gold each sack contains. Instead I will ask you to show that the treasure hunter can always appease the guardians.

 

A more mathematical way of phrasing the problem: given a set of 14 integers S = {n1, n2, ..., n14}, where 0 < ni < 1000 for all i, show that there must exist two disjoint non-empty subsets of S with the same sum.


Chase on an endless road

30 July 2013 - 12:01 PM

You and your friend are standing on a road which goes on forever in both directions. Your friend suggests a game of chase. Since you are faster, your friend will get a head start. Also, to make it more interesting than a simple Achilles vs Tortoise race, you will not know in which direction your friend takes off.

  • You move at constant speed x>1 m/s and your friend moves at constant speed 1 m/s.
  • Your friend gets a 1 hour head start.
  • Visibility is limited so you will not be able to see your friend from a distance.
  • You must not stray from the road, same goes for your friend.

Can you catch up to your friend? Does the answer depend on your speed x, and if so, what is the minimum x for which you can catch up? What is your strategy?