Euler problems/111 to 120

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Revision as of 22:22, 15 August 2007

1 Problem 111

Search for 10-digit primes containing the maximum number of repeated digits.

Solution:

problem_111 = undefined

2 Problem 112

Investigating the density of "bouncy" numbers.

Solution:

problem_112 = undefined

3 Problem 113

How many numbers below a googol (10100) are not "bouncy"?

Solution:

import Array
 
mkArray b f = listArray b $ map f (range b)
 
digits = 100
 
inc = mkArray ((1, 0), (digits, 9)) ninc
dec = mkArray ((1, 0), (digits, 9)) ndec
 
ninc (1, _) = 1
ninc (l, d) = sum [inc ! (l-1, i) | i <- [d..9]]
 
ndec (1, _) = 1
ndec (l, d) = sum [dec ! (l-1, i) | i <- [0..d]]
 
problem_113 = sum [inc ! i | i <- range ((digits, 0), (digits, 9))]
               + sum [dec ! i | i <- range ((1, 1), (digits, 9))]
               - digits*9 -- numbers like 11111 are counted in both inc and dec 
               - 1 -- 0 is included in the increasing numbers

Note: inc and dec contain the same data, but it seems clearer to duplicate them.

4 Problem 114

Investigating the number of ways to fill a row with separated blocks that are at least three units long.

Solution:

problem_114 = undefined

5 Problem 115

Finding a generalisation for the number of ways to fill a row with separated blocks.

Solution:

problem_115 = undefined

6 Problem 116

Investigating the number of ways of replacing square tiles with one of three coloured tiles.

Solution:

problem_116 = undefined

7 Problem 117

Investigating the number of ways of tiling a row using different-sized tiles.

Solution:

problem_117 = undefined

8 Problem 118

Exploring the number of ways in which sets containing prime elements can be made.

Solution:

problem_118 = undefined

9 Problem 119

Investigating the numbers which are equal to sum of their digits raised to some power.

Solution:

problem_119 = undefined

10 Problem 120

Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.

Solution:

problem_120 = undefined

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