# Problem 63 Powerful digit counts

## Problem 63: Powerful digit counts

https://projecteuler.net/problem=63

The 5-digit number, 16807=75, is also a fifth power. Similarly, the 9-digit number, 134217728=89, is a ninth power.

How many n-digit positive integers exist which are also an nth power?

## 分析

16807 是 5 位数，也是 7 的 5 次方
134217728 是 9[……]

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# Problem 62 Cubic permutations

## Problem 62: Cubic permutations

https://projecteuler.net/problem=62

The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the sma[……]

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# Problem 61 Cyclical figurate numbers

## Problem 61: Cyclical figurate numbers

https://projecteuler.net/problem=61

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, …
Square[……]

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# Problem 60 Prime pair sets

## Problem 60: Prime pair sets

https://projecteuler.net/problem=60

The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four prim[……]

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# Problem 59 XOR decryption

## Problem 59: XOR decryption

https://projecteuler.net/problem=59

Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.

A modern encrypti[……]

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# Problem 58 Spiral primes

## Problem 58: Spiral primes

https://projecteuler.net/problem=58

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed. It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8[……]

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# Problem 57: Square root convergents

## Problem 57: Square root convergents

https://projecteuler.net/problem=57 ## 分析

第 1 次扩展结果: fraction = 1 + 1/2

# Problem 56 Powerful digit sum

## Problem 56: Powerful digit sum

https://projecteuler.net/problem=56

A googol (10100) is a massive number: one followed by one-hundred zeros; 100100 is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1.

Considering natura[……]

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# Problem 55 Lychrel numbers

## Problem 55: Lychrel numbers

https://projecteuler.net/problem=55

If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.

Not all numbers produce palindromes so quickly. For example,

349 + 943 = 1292,
1292 + 2921 = 4213
4213 + 3124 = 7337

That is, 349 took three iterations to arrive at[……]

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# Problem 54 Poker hands

## Problem 54: Poker hands

https://projecteuler.net/problem=54 ## 分析

• 高牌，即大散牌(High Card): Highest value card.
• 一对(One Pair)
• 两对(Two Pairs)
• 三条(Three of a Kind)
• 顺子(Straight)
• 同花(Flush)
• 葫芦，即三条+一对(Full House)
• 四条(Four of a Kind)
• 同花顺(Straight Flush)
• 皇家[……]

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# Problem 53 Combinatoric selections

## Problem 53: Combinatoric selections

https://projecteuler.net/problem=53 ## 分析

C(n,r) = n! / r!(n-r)!，计算过程中，可以先做约分，再计算最终结果。假设 max = max(r,n-r). 可以先约去 max!。 m[……]

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# Problem 52 Permuted multiples

## Problem 52: Permuted multiples

https://projecteuler.net/problem=52

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.

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

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# Problem 51 Prime digit replacements

## Problem 51: Prime digit replacements

https://projecteuler.net/problem=51

By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit[……]

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# Problem 50 Consecutive prime sum

## Problem 50: Consecutive prime sum

https://projecteuler.net/problem=50

The prime 41, can be written as the sum of six consecutive primes:

41 = 2 + 3 + 5 + 7 + 11 + 13
This is the longest sum of consecutive primes that adds to a prime below one-hundred.

The longest sum of consecutive primes below on[……]

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# Problem 44 Pentagon numbers

## Problem 44: Pentagon numbers

https://projecteuler.net/problem=44

Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, …

It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48[……]

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