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 Extreme Sum Settlement (Posted on 2015-09-28)
Determine the total number of 8-digit base ten positive integers such that the sum of the four leftmost digits is equal to precisely 15 more than the sum of four rightmost digits.

*** Assume non leading zero for each of the numbers.

 No Solution Yet Submitted by K Sengupta No Rating

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 the computer doesn't tire of writing the numbers | Comment 2 of 4 |
The first four digits must not allow leading zeros, but any digit including zero may be in the 5th position, so the last four digits may begin with zero.

`                        Ways for given s.o.d. among 4 digits sum of digits      0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17 w/o leading zeros  0   1   4  10  20  35  56  84 120 165 219 279 342 405 465 519 564 597 with leading zeros 1   4  10  20  35  56  84 120 165 220 282 348 415 480 540 592 633 660                         18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36                       615 615 597 564 519 465 405 342 279 219 165 120  84  56  35  20  10   4   1                       670 660 633 592 540 480 415 348 282 220 165 120  84  56  35  20  10   4   1                       The ways for a full 8-digit number is the number of ways for each part separately, multiplied together:   1 * 519 =   519   4 * 564 =  2256  10 * 597 =  5970  20 * 615 = 12300  35 * 615 = 21525  56 * 597 = 33432  84 * 564 = 47376 120 * 519 = 62280 165 * 465 = 76725 220 * 405 = 89100 282 * 342 = 96444 348 * 279 = 97092 415 * 219 = 90885 480 * 165 = 79200 540 * 120 = 64800 592 *  84 = 49728 633 *  56 = 35448 660 *  35 = 23100 670 *  20 = 13400 660 *  10 =  6600 633 *   4 =  2532 592 *   1 =   592total       911304 such 8-digit integers.`

DefDbl A-Z
Dim crlf\$, nlzero(36), lzero(36)

Private Sub Form_Load()
Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For a = 0 To 9
For b = 0 To 9
For c = 0 To 9
For d = 0 To 9
t = a + b + c + d
If a <> 0 Then nlzero(t) = nlzero(t) + 1
lzero(t) = lzero(t) + 1
Next
Next
Next
Next

For i = 0 To 36
Text1.Text = Text1.Text & mform(i, "###0")
Next
Text1.Text = Text1.Text & crlf
For i = 0 To 36
Text1.Text = Text1.Text & mform(nlzero(i), "###0")
Next
Text1.Text = Text1.Text & crlf
For i = 0 To 36
Text1.Text = Text1.Text & mform(lzero(i), "###0")
Next
Text1.Text = Text1.Text & crlf

For low = 0 To 36 - 15
Text1.Text = Text1.Text & mform(lzero(low), "###0") & " *" & mform(nlzero(low + 15), "###0") & " =" & mform(lzero(low) * nlzero(low + 15), "#####0") & crlf
totnumber = totnumber + lzero(low) * nlzero(low + 15)
Next

Text1.Text = Text1.Text & totnumber & 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

 Posted by Charlie on 2015-09-28 15:56:47
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