# Difference between revisions of "Euler problems/111 to 120"

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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_111 = undefined |
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+ | import Control.Monad (replicateM) |
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+ | |||

+ | -- All ways of interspersing n copies of x into a list |
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+ | intr :: Int -> a -> [a] -> [[a]] |
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+ | intr 0 _ y = [y] |
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+ | intr n x (y:ys) = concat |
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+ | [map ((replicate i x ++) . (y :)) $ intr (n-i) x ys |
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+ | | i <- [0..n]] |
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+ | intr n x _ = [replicate n x] |
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+ | |||

+ | -- All 10-digit primes containing the maximal number of the digit d |
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+ | maxDigits :: Char -> [Integer] |
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+ | maxDigits d = head $ dropWhile null |
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+ | [filter isPrime $ map read $ filter ((/='0') . head) $ |
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+ | concatMap (intr (10-n) d) $ |
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+ | replicateM n $ delete d "0123456789" |
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+ | | n <- [1..9]] |
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+ | |||

+ | problem_111 = sum $ concatMap maxDigits "0123456789" |
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</haskell> |
</haskell> |
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## Revision as of 18:54, 12 November 2007

## Contents

## Problem 111

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

Solution:

```
import Control.Monad (replicateM)
-- All ways of interspersing n copies of x into a list
intr :: Int -> a -> [a] -> [[a]]
intr 0 _ y = [y]
intr n x (y:ys) = concat
[map ((replicate i x ++) . (y :)) $ intr (n-i) x ys
| i <- [0..n]]
intr n x _ = [replicate n x]
-- All 10-digit primes containing the maximal number of the digit d
maxDigits :: Char -> [Integer]
maxDigits d = head $ dropWhile null
[filter isPrime $ map read $ filter ((/='0') . head) $
concatMap (intr (10-n) d) $
replicateM n $ delete d "0123456789"
| n <- [1..9]]
problem_111 = sum $ concatMap maxDigits "0123456789"
```

## Problem 112

Investigating the density of "bouncy" numbers.

Solution:

```
problem_112 = undefined
```

## 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.

## 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
```

## Problem 115

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

Solution:

```
problem_115 = undefined
```

## Problem 116

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

Solution:

```
problem_116 = undefined
```

## Problem 117

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

Solution:

```
problem_117 = undefined
```

## Problem 118

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

Solution:

```
problem_118 = undefined
```

## Problem 119

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

Solution:

```
problem_119 = undefined
```

## Problem 120

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

Solution:

```
problem_120 = undefined
```