Math Is Fun Forum

Here is one promising algorithm which does not work:

for each Number
   for each Row
        take a random Column with an empty cell in the Row
        check that all cells above it do not contain Number
        if this column already occupied by this Number - find a new Column and repeat.
        once non-occupied row is found write Number in the cell
   next Row
next Number

Problem here is that in some cases this algorithm can stuck:
Working with 4x4 board:
For number 1: put it in 1, 2, 3, 4 positions in rows
For number 2: put it in 2, 3, 1, .... No where to put 2 in fourth row

12..
.12.
2.1.
...1

Any ideas why this promising algorithm fails and how to improve it to prevent blocks like this?

Since I do not understand your answer, the next question would be: "What books should I read?"

And what program do you use for this formula conversion?

I am a little shaky on this topic myself, but...
Your R is from set {A, B, C} to set {1, 2, 3}. So the domain would be a union of those two sets: {A, B, C, 1, 2, 3}

Imagine that each one shift produces a hundred items. The first shift produced 100 total and 1%=one item from the produced hundred is defective. The second shift: 100 total and 2%=2 items are defective. Same for the third shift. So in total, during the day, all three shifts produced 100*3=300 items. From this 300, there were 1+2+5=8 defective items. 8 items of 300 is 2.(6)%
Or, since all shifts are equal, you can simply take the average of three, but this solution would work only for equal shifts.

Yes. I work with Rosen's book "Discrete Mathematics and Its Applications"; in it the wording of the problem is "to solve the recurrence". But yes, the task is actually to convert a recurrence form into a normal function.

I see one difference in our approaches: you went backward (or forward?) starting from the

Ok... I finished the chapter 5.5 and at the end of it I found the description of this formula. So I tried to do a manual calculation and received -41 (negative 41)... After a while I realized that I made a mistake in writing formula down and replaced the i with j in one place (powered -1 before the binom).
But now I have an interesting equality:

To get rid of implication arrow you can use the conversion rule

And the second step is using De Morgans law

soroban, Sorry, you made two mistakes:

And here you forgot the very first case with all objects together. So the final summation should be