You are on the Home/Other Tutorials/Project Euler/Problem 40 page
Web This Site

Project Euler - Problem 40

More about Project Euler.

Problem description

An irrational decimal fraction is created by concatenating the positive integers:


It can be seen that the 12^(th) digit of the fractional part is 1.

If d_(n) represents the n^(th) digit of the fractional part, find the value of the following expression.

d_(1) x d_(10) x d_(100) x d_(1000) x d_(10000) x d_(100000) x d_(1000000


The brute force VBA approach below took 200 seconds on my laptop.  Of course, I could run it on the much faster desktop in under a minute but here's how to do it by hand:

There are 9 numbers (1 through 9) with 1 digit each.  In total they account for 9 digits.

There are 90 numbers (10 through 99) with 2 digits each.  In total they account for 180 digits.  Numbers with 1 or 2 digits account for 189 digits.

There are 900 numbers (100 through 999) with 3 digits each.  In total they account for 2700 digits.  Numbers with 1, 2, or 3 digits account for 2889 digits.

The table below lists the range of numbers required to get to 1,000,000 digits.

Now, in some other range, create the following:

In I15:I20 enter the formulas

and copy them to J:O.  The required result is the product of I20:O20.

Option Explicit
Option Base 0

Sub Euler040()
    Dim IrrNbr As String, I As Long, ProcTime As Single
    I = 1
    ProcTime = Timer
    Do While Len(IrrNbr) < 10 ^ 6
        IrrNbr = IrrNbr & CStr(I): I = I + 1
    Dim MultRslt As Integer
    MultRslt = 1
    For I = 0 To 6
        MultRslt = MultRslt * CInt(Mid(IrrNbr, 10 ^ I, 1))
        Next I
    Debug.Print MultRslt, Timer - ProcTime
    End Sub