Kings all over the game

There are 2 such arrangements. Suppose that we have an arrangement satisfying the problem conditions. Divide the board into 2 x 2 pieces; we call these pieces blocks. Each block can contain not more than one king (otherwise the 2 kings would attack each other); hence, by the pigeonhole principle each block must contain exactly one king. Now assign to each block a letter T or B if a king is placed in its top or bottom half, respectively. Similarly, assign to each block a letter L or R if a king stands in its left or right half. So we define T-blocks, B-blocks, L-blocks and R-blocks. We also combine the letters; we call a block a TL-block if it is simultaneously T-block and L-block. Similarly we define TR-blocks, BL-blocks, BR-blocks. The arrangement of blocks determines uniquely the arrangement of kings; so in the rest of the solution we consider the 50 x 50 system of blocks (see Fig. 1). We identify the blocks by their coordinate pairs; the pair (i, j), where 1 ≤ i, j ≤ 50, refers to the jth block in the ith row (or the ith block in the jth column). The upper-left block is (1, 1). The system of blocks has the following properties : (i) If (i, j) is a B-block the (i+1, j) is a B-block: otherwise the kings in these 2 blocks can take each other.
Similarly: if (i, j) is a T-block then (i-1, j) is a T-block; if (i,j) is a L-block the (i, j-1) is a L-block; if (i, j) is a R-
block then (i, j+1) is a R-block. (ii)Each column contains exactly 25 L-blocks and 25 R-blocks, and each row contains exactly 25 T-blocks and
25 R-blocks. In particular, the total number of L-blocks (or R-blocks, or T-blocks, or B-blocks) is equal to 25 . 50=1250. Consider any B-blocks of the form (i, j). By (i), all blocks in the jth column are B-blocks; so we call such a column B-column. By (ii), we have 25 B-blocks in the first row, so we obtain 25 B-columns. These 25 B-columns contains 1250 B-blocks, hence all blocks in the remaining columns arwe T-blocks, and we obtain 25 T-columns. Similarly, there are exactly 25 L-rows and exactly 25 R-rows. Now consider an arbitrary pair of a T-column and a neighbouring B-column (columns with numbers j and j+1). Case 1: Suppose that the jth column is a T-column, and the (j+1)th column is a B-column. Consider some index i such that the ith row is an L-row; then (i, j+1) is a BL-block. Therefore, (i+1, j) cannot be a TR-block, hence (i+1, J) is a TL-block, thus the (i+1)th row is a L-row. Now, choosing the ith row to be the topmost L-row, we successively obtain that all rows from the ith row to the 50th row are L-rows. Since we have exactly 25 L-rows, it follows that the rows from the 1st to the 25th are R-rows, and the rows from the 26th to the 50th are L-rows. Now consider the neighbouring R-row and L-row (that are the rows with numbers 25 and 26). Replacing in the previous reasoning rows by columns and vice versa, the columns from the 1st to the 25th are T-columns, and the columns from the 26th to the 50th are B-columns. So we have a unique arrangement of blocks that leads to the arrangement of kings satisfying the condition of the problem. Case 2: Suppose that the jth column is a B-column, and the (j+1)th column is a T-column. Repeating the arguments from Case 1, we obtain that the rows from the 1st to the 25th are B-columns (and all columns are T-columns), so we find exactly one more arrangement of kings.