# Fill two dimensional list by meeting given conditions

Suppose you have a 2D array - a list that exits in both x and y axis - which is 4 x 4, and you need to fill the table by putting numbers in it obeying following restrictions.

1. Each cell is able to store a number that can be calculated according to its index. (Indexes start from [0,0])

If a cell's index is [i, j] it can store a number calculated as one of the following ways.

1. i x j + i - j

2. i x j + j - i

3. i ^ j

4. j ^ i

2. Each cell can carry at most multiplication of its adjacent cells. (horizonal and vertical)

3. Each cell should be colored in a way all odds are of same color and all evens are of same color and two same colored cell cannot stand side by side vertically or horizontally.

• How do you define $0^0$? Mar 29, 2019 at 9:59
• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. If not, some responses to the answerers to help steer them in the right direction would be helpful.
– Rubio
Apr 2, 2019 at 18:11

It turns out that this is

Impossible

Reasoning

The cell with index $$[1,1]$$ must contain the entry $$1\times 1 + 1 - 1 = 1^1 = 1$$.

Now consider the cell with index $$[0,1]$$ which is adjacent to $$[1,1]$$. This must contain either
(i) $$0 \times 1 + 0 - 1 = -1$$,
(ii) $$0 \times 1 + 1 - 0 = 1$$
(iii) $$0^1 = 0$$
(iv) $$1^0 = 1$$

Now, (i), (ii) and (iv) are odd which would break rule 3.
However if we are in case (iii) then we break rule 2, since the cell with index $$[1,1]$$ can carry, at most, multiplication of its adjacent cells, which would be $$0$$ in this case.

• Actually it may be possible due to the fact that (according to the rules) rot13(rnpu pryy VF NOYR (be PNA) pbagnva n ahzore, ohg abg arprffnevyl FUBHYQ gb qb fb. Fbzr pryyf znl pbagnva ab ahzoref ng nyy (naq fb, abg orvat terngre guna nal bgure ahzoref, yvxr AnA va cebtenzzvat)) Mar 29, 2019 at 10:20
• @trolley813 That's quite an interesting take. Wouldn't it contravene the sentence "...you need to fill the table by putting numbers in it obeying following restrictions." in the opening paragraph? Mar 29, 2019 at 10:26
• maybe, but you can fill the table partially. I'm now writing my "solution". Mar 29, 2019 at 10:31

My "solution" (a lateral thinking answer)

__0 NaN 1 NaN
NaN 1 NaN 1
__1 NaN 4 NaN
NaN 1 NaN 9

Explanation

NaN meaning not-a-number (according to the rules, a cell does not has to contain a number, but rather can do so). All other rules are obviously complied (if a cell contains a number, it is of the 4 permitted ones; NaNs (and their products) are never greater than any number; you should color all "odd" cells blue, "evens" green and NaNs grey, since the latter are neither even nor odd).