Hi everyone, :)
Suppose we have four numbers which share one digit in common. Then the only way to arrange those four numbers with one number at the first row-first column position is,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]x[/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]x[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]x[/TD]
[TD][/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD="align: center"]x[/TD]
[TD][/TD]
[/TR]
[/TABLE]
Now suppose we are given another four numbers which have a digit in common that we have to place in the table such that one number among those four numbers should be second row, second column position. Then,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]x[/TD]
[TD="align: center"][/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"][/TD]
[/TR]
[TR]
[TD="align: center"][/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"][/TD]
[TD="align: center"]x
[/TD]
[/TR]
[TR]
[TD="align: center"][/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"][/TD]
[TD="align: center"]y
[/TD]
[/TR]
[TR]
[TD="align: center"]y
[/TD]
[TD="align: center"][/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"][/TD]
[/TR]
[/TABLE]
Similarly given a third quadruplet of numbers...
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]x
[/TD]
[TD="align: center"][/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"]z
[/TD]
[/TR]
[TR]
[TD="align: center"]z
[/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"][/TD]
[TD="align: center"]x
[/TD]
[/TR]
[TR]
[TD="align: center"][/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"]z
[/TD]
[TD="align: center"]y
[/TD]
[/TR]
[TR]
[TD="align: center"]y
[/TD]
[TD="align: center"]z
[/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"][/TD]
[/TR]
[/TABLE]
And the final four numbers...
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]x
[/TD]
[TD="align: center"]m
[/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"]z
[/TD]
[/TR]
[TR]
[TD="align: center"]z
[/TD]
[TD="align: center"]y
[/TD]
[TD="align: center"]m
[/TD]
[TD="align: center"]x
[/TD]
[/TR]
[TR]
[TD="align: center"]m
[/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"]z
[/TD]
[TD="align: center"]y
[/TD]
[/TR]
[TR]
[TD="align: center"]y
[/TD]
[TD="align: center"]z
[/TD]
[TD="align: center"]x
[/TD]
[TD="align: center"]m
[/TD]
[/TR]
[/TABLE]
In our problem the situation is similar. We can arrange the given numbers into four groups:
Numbers with the digit 1 in common: 10,12,13,21 (x-group)
Numbers with the digit 4 in common: 34,40,47,64 (y-group)
Numbers with the digit 5 in common: 50,53,57,65 (z-group)
Numbers with the digit 8 in common: 38,78,89,98 (m-group)
Now let us begin the arrangement.
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]10[/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]12[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]13[/TD]
[TD][/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD="align: center"]21[/TD]
[TD][/TD]
[/TR]
[/TABLE]
Now we move to the second group. Note that the number 34 which is in the second set, can only be placed in the following position,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]10[/TD]
[TD="align: center"][/TD]
[TD="align: center"]34[/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]12[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]13[/TD]
[TD][/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD="align: center"]21[/TD]
[TD][/TD]
[/TR]
[/TABLE]
Similarly the number 40 is forced to place as follows,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]10[/TD]
[TD="align: center"][/TD]
[TD="align: center"]34[/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]12[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]13[/TD]
[TD][/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD="align: center"]21[/TD]
[TD][/TD]
[/TR]
[/TABLE]
Now consider the number 50 in the third group. The only position it can take is,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]10[/TD]
[TD="align: center"][/TD]
[TD="align: center"]34[/TD]
[TD][/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]12[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]13[/TD]
[TD][/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD][/TD]
[TD]50[/TD]
[TD="align: center"]21[/TD]
[TD][/TD]
[/TR]
[/TABLE]
Then for the number 53 the only option is,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD="align: center"]10[/TD]
[TD="align: center"][/TD]
[TD="align: center"]34[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD][/TD]
[TD][/TD]
[TD="align: center"]12[/TD]
[/TR]
[TR]
[TD][/TD]
[TD="align: center"]13[/TD]
[TD][/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD][/TD]
[TD]50[/TD]
[TD="align: center"]21[/TD]
[TD][/TD]
[/TR]
[/TABLE]
As usual 38 has only one place to go,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD]10[/TD]
[TD][/TD]
[TD]34[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD="align: center"][/TD]
[TD][/TD]
[TD]12[/TD]
[/TR]
[TR]
[TD="align: center"][/TD]
[TD]13[/TD]
[TD][/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD="align: center"][/TD]
[TD]50[/TD]
[TD]21[/TD]
[TD]38[/TD]
[/TR]
[/TABLE]
The remaining numbers are,
Numbers with the digit 4 in common: 47,64 (y-group)
Numbers with the digit 5 in common: 57,65 (z-group)
Numbers with the digit 8 in common: 78,89,98 (m-group)
So we have two possible y-values that can fill the second row-second column box. If we use 47 we get two tables;
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD]10[/TD]
[TD]89[/TD]
[TD]34[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD="align: center"]47[/TD]
[TD]98[/TD]
[TD]12[/TD]
[/TR]
[TR]
[TD="align: center"]78[/TD]
[TD]13[/TD]
[TD]65[/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD="align: center"]64[/TD]
[TD]50[/TD]
[TD]21[/TD]
[TD]38[/TD]
[/TR]
[/TABLE]
and,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD]10[/TD]
[TD]98[/TD]
[TD]34[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD="align: center"]47[/TD]
[TD]89[/TD]
[TD]12[/TD]
[/TR]
[TR]
[TD="align: center"]78[/TD]
[TD]13[/TD]
[TD]65[/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD="align: center"]64[/TD]
[TD]50[/TD]
[TD]21[/TD]
[TD]38[/TD]
[/TR]
[/TABLE]
If we use 64 to fill the second row-second column box we have another two possibilities,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD]10[/TD]
[TD]78[/TD]
[TD]34[/TD]
[TD]65[/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD="align: center"]64[/TD]
[TD]98[/TD]
[TD]12[/TD]
[/TR]
[TR]
[TD="align: center"]89[/TD]
[TD]13[/TD]
[TD]57[/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD="align: center"]47[/TD]
[TD]50[/TD]
[TD]21[/TD]
[TD]38[/TD]
[/TR]
[/TABLE]
and,
[TABLE="class: grid, width: 100, align: center"]
[TR]
[TD]10[/TD]
[TD]78[/TD]
[TD]34[/TD]
[TD]65[/TD]
[/TR]
[TR]
[TD]53[/TD]
[TD="align: center"]64[/TD]
[TD]89[/TD]
[TD]12[/TD]
[/TR]
[TR]
[TD="align: center"]98[/TD]
[TD]13[/TD]
[TD]57[/TD]
[TD]40[/TD]
[/TR]
[TR]
[TD="align: center"]47[/TD]
[TD]50[/TD]
[TD]21[/TD]
[TD]38[/TD]
[/TR]
[/TABLE]
The whole thing would change if I had used a different element than 10 in the first box. There are \(4!\) possible arrangements for the initial table where only the x-group elements are listed, so I feel that this problem has quite a number of solutions. :)
Kind Regards,
Sudharaka.