Project Euler

[ホーム](https://redmine.noppi.jp) - [[Wiki|Project Euler]]

# [[Problem 55]]

## Lychrel Numbers

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

<p>Not all numbers produce palindromes so quickly. For example,</p>

\begin{align}

349 + 943 &= 1292\\

1292 + 2921 &= 4213\\

4213 + 3124 &= 7337

\end{align}

That is, $349$ took three iterations to arrive at a palindrome.

Although no one has proved it yet, it is thought that some numbers, like $196$, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, $10677$ is the first number to be shown to require over fifty iterations before producing a palindrome: $4668731596684224866951378664$ ($53$ iterations, $28$-digits).

Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is $4994$.

How many Lychrel numbers are there below ten-thousand?

NOTE: Wording was modified slightly on 24 April 2007 to emphasise the theoretical nature of Lychrel numbers.

## Lychrel数

47とその反転を足し合わせると, 47 + 74 = 121となり, 回文数になる.

全ての数が素早く回文数になるわけではない. 349を考えよう,

1. 349 + 943 = 1292,

1. 1292 + 2921 = 4213

1. 4213 + 3124 = 7337

349は, 3回の操作を経て回文数になる.

まだ証明はされていないが, 196のようないくつかの数字は回文数にならないと考えられている. 反転したものを足すという操作を経ても回文数にならないものをLychrel数と呼ぶ. 先のような数の理論的な性質により, またこの問題の目的のために, Lychrel数で無いと証明されていない数はLychrel数だと仮定する.

更に, 10000未満の数については，常に以下のどちらか一方が成り立つと仮定してよい.

1. 50回未満の操作で回文数になる

1. まだ誰も回文数まで到達していない

実際, 10677が50回以上の操作を必要とする最初の数である: 4668731596684224866951378664 (53回の操作で28桁のこの回文数になる).

驚くべきことに, 回文数かつLychrel数であるものが存在する. 最初の数は4994である.

10000未満のLychrel数の個数を答えよ.

注: 2007/04/24にLychrel数の理論的な性質を強調するために文面が修正された.

```scheme

(import (scheme base)

        (gauche base))

(define (rotation-number num)

  (assume (exact-integer? num))

  (assume (not (negative? num)))

  (if (zero? num)

    0

    (let loop ([rest num] [result 0])

      (if (zero? rest)

        result

        (loop (div rest 10)

              (+ (* result 10)

                 (mod rest 10)))))))

(define (palindrome? num)

  (assume (exact-integer? num))

  (assume (not (negative? num)))

  (= num

     (rotation-number num)))

(define (lychrel? num)

  (assume (exact-integer? num))

  (assume (not (negative? num)))

  (let loop ([index 0] [current num])

    (if (<= 50 index)

      #t

      (let ([next (+ current

                     (rotation-number current))])

        (if (palindrome? next)

          #f

          (loop (+ index 1) next))))))

(define answer-55

  (length

    (filter lychrel?

            (iota 10000))))

(format #t "55: ~d~%" answer-55)

```