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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 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.

Using the Large Number Arithmetic module, specifically, the LargeAdd routine it was easy to directly implement the problem description into code. Since a requirement towards getting the correct solution was to compute the actual palindrome, I took the time to list them.

Function IsPalindrome(ByVal X As String) As Boolean IsPalindrome = X = StrReverse(X) End Function Sub Euler055() Dim I As Integer, aNbr As String, Palindromes() As String, _ PalindromeCount As Integer, _ TimeIt As Single TimeIt = Timer ReDim Palindromes(9998) For I = 1 To 9999 aNbr = CStr(I) Dim RetryCount As Integer, EndLoop As Boolean RetryCount = 0: EndLoop = False Do aNbr = LargeAdd(aNbr, StrReverse(aNbr)) If IsPalindrome(aNbr) Then Palindromes(PalindromeCount) = aNbr PalindromeCount = PalindromeCount + 1 EndLoop = True ElseIf RetryCount >= 50 Then EndLoop = True Else RetryCount = RetryCount + 1 End If Loop Until EndLoop Next I Debug.Print PalindromeCount, Timer - TimeIt ReDim Preserve Palindromes(PalindromeCount - 1) ActiveSheet.Cells(1, 1).Resize(PalindromeCount, 1).Value = _ Application.WorksheetFunction.Transpose(Palindromes) End Sub