Prove that no matter how each cell of a 5 x 41 table is filled with a 0 or 1, one can choose 3 rows and 3 columns which intersect in 9 cells filled with identical numbers.
Prove that 41 is the lowest possible n for 5 x n table; i.e., the statement is not true for a 5 x 40 table.
Source: Colorado math contest.
(In reply to
Example needed by broll)
first example below: rows 1, 15 and 29; columns 1, 2 and 4:
0 0 1 0 1
0 0 1 0 1
0 0 1 0 1
those columns in those rows are all zeros
likewise for the other sets listed below
rows columns
1 15 29 1 2 4
2 16 30 1 3 5
3 17 31 2 3 5
4 14 18 2 3 5
4 14 28 2 3 5
4 14 32 2 3 5
4 18 28 2 3 5
4 18 32 2 3 5
4 28 32 2 3 5
5 19 33 1 3 5
6 20 34 1 2 4
7 11 21 2 3 4
7 11 25 2 3 4
7 11 35 2 3 4
7 11 39 2 3 4
7 21 25 2 3 4
7 21 35 2 3 4
7 21 39 2 3 4
7 25 35 2 3 4
7 25 39 2 3 4
7 35 39 2 3 4
8 22 36 1 2 5
9 23 37 1 3 4
10 24 38 2 3 4
11 21 25 2 3 4
11 21 35 2 3 4
11 21 39 2 3 4
11 25 35 2 3 4
11 25 39 2 3 4
11 35 39 2 3 4
12 26 40 1 3 4
13 27 41 1 2 5
14 18 28 2 3 5
14 18 32 2 3 5
14 28 32 2 3 5
18 28 32 2 3 5
21 25 35 2 3 4
21 25 39 2 3 4
21 35 39 2 3 4
25 35 39 2 3 4
DefDbl A-Z
Dim crlf$, grid(41, 5)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
Open "tables peculiarity.txt" For Input As #1
For i = 1 To 41
Line Input #1, s$
For p = 2 To 10 Step 2
grid(i, p / 2) = Val(Mid(s, p, 1))
Next
Next
Close
For r1 = 1 To 39
For r2 = r1 + 1 To 40
For r3 = r2 + 1 To 41
For c1 = 1 To 3
For c2 = c1 + 1 To 4
For c3 = c2 + 1 To 5
DoEvents
If grid(r1, c2) = grid(r1, c1) Then
If grid(r1, c3) = grid(r1, c1) Then
If grid(r2, c1) = grid(r1, c1) Then
If grid(r2, c2) = grid(r1, c1) Then
If grid(r2, c3) = grid(r1, c1) Then
If grid(r3, c1) = grid(r1, c1) Then
If grid(r3, c2) = grid(r1, c1) Then
If grid(r3, c3) = grid(r1, c1) Then
Text1.Text = Text1.Text & r1 & Str(r2) & Str(r3) & " " & Str(c1) & Str(c2) & Str(c3) & crlf
End If
End If
End If
End If
End If
End If
End If
End If
Next
Next
Next
Next
Next
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2015-11-30 08:24:58 |
re: Example needed
