For how many integers having between 1 and 10 digits (base 10) are all of their digits when read from left to right monotonically increasing? In other words, every digit is less than or equal to all of those to its right. For example, 244467889 is one of them, and 0 is another, but there are more.
How many 1-digit numbers satisfy the condition? Categorized by their last (in this case, only) digit, each has one satisfactory example:
last digit: 0 1 2 3 4 5 6 7 8 9
number: 1 1 1 1 1 1 1 1 1 1
For 2-digit numbers, any 1-digit monotonous number ending with the same or lower digit can be extended by the one digit:
last digit: 0 1 2 3 4 5 6 7 8 9
number: 1 2 3 4 5 6 7 8 9 10
But these "2-digit numbers" include numbers that start with zero, and so the total of the numbers shown, 55, is really the total of 1- and 2-digit numbers that satisfy the criterion.
For 3-digit numbers, any 2-digit monotonous number ending with the same or lower digit can be extended by the one digit:
last digit: 0 1 2 3 4 5 6 7 8 9
number: 1 3 6 10 15 21 28 36 45 55
Again, of course, the total of these, 220, includes 1- and 2-digit monotonous numbers as well as the 3-digit variety.
Of course I'm tired of completing this by hand and then adding all the numbers. I'll let the computer do it.
max
digits line total
4 1 4 10 20 35 56 84 120 165 220 715
5 1 5 15 35 70 126 210 330 495 715 2002
6 1 6 21 56 126 252 462 792 1287 2002 5005
7 1 7 28 84 210 462 924 1716 3003 5005 11440
8 1 8 36 120 330 792 1716 3432 6435 11440 24310
9 1 9 45 165 495 1287 3003 6435 12870 24310 48620
10 1 10 55 220 715 2002 5005 11440 24310 48620 92378
So from 1- to 10-digit numbers, 92,378 are monotonous.
DefDbl A-Z
Dim crlf$, grid(10, 10)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For lastdig = 0 To 9
grid(1, lastdig) = 1
Next
For numdigits = 2 To 10
Text1.Text = Text1.Text & mform(numdigits, "##0")
tot = 0
For lastdig = 0 To 9
t = 0
For i = 0 To lastdig
t = t + grid(numdigits - 1, i)
Next
grid(numdigits, lastdig) = t
tot = tot + t
Text1.Text = Text1.Text & mform(t, "#####0")
Next
Text1.Text = Text1.Text & " " & mform(tot, "##########0") & crlf
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Note, the program also verifies the lines for 2- and 3-digit maximum lengths.
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Posted by Charlie
on 2015-12-30 13:51:37 |
solution, starting out with analytic
