# Euler problems/51 to 60

(Difference between revisions)

## 1 Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

`problem_51 = undefined`

## 2 Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

`problem_52 = undefined`

## 3 Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

`problem_53 = undefined`

## 4 Problem 54

How many hands did player one win in the game of poker?

Solution:

`problem_54 = undefined`

## 5 Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

```problem_55 = length \$ filter isLychrel [1..9999]
where isLychrel n = all notPalindrome (take 50 (tail (iterate revadd n)))
notPalindrome s = (show s) /= reverse (show s)
revadd n = n + rev n
where rev n = read (reverse (show n))```

## 6 Problem 56

Considering natural numbers of the form, ab, finding the maximum digital sum.

Solution:

```problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]]
where dsum 0 = 0
dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )```

## 7 Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

`problem_57 = undefined`

## 8 Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

`problem_58 = undefined`

## 9 Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

`problem_59 = undefined`

## 10 Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

`problem_60 = undefined`
The text passage contains a list of mathematical problems and their corresponding solutions, where some solutions are defined and others are not. Each problem is a distinct mathematical question.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Euler problems/51 to 60

(Difference between revisions)

## 1 Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

`problem_51 = undefined`

## 2 Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

`problem_52 = undefined`

## 3 Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

`problem_53 = undefined`

## 4 Problem 54

How many hands did player one win in the game of poker?

Solution:

`problem_54 = undefined`

## 5 Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

```problem_55 = length \$ filter isLychrel [1..9999]
where isLychrel n = all notPalindrome (take 50 (tail (iterate revadd n)))
notPalindrome s = (show s) /= reverse (show s)
revadd n = n + rev n
where rev n = read (reverse (show n))```

## 6 Problem 56

Considering natural numbers of the form, ab, finding the maximum digital sum.

Solution:

```problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]]
where dsum 0 = 0
dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )```

## 7 Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

`problem_57 = undefined`

## 8 Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

`problem_58 = undefined`

## 9 Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

`problem_59 = undefined`

## 10 Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

`problem_60 = undefined`
user
The text passage contains a list of mathematical problems and their corresponding solutions, where some solutions are defined and others are not. Each problem is a distinct mathematical question.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS