I found the instructions for the Sam Loyd - Columbus problem vague and misleading. The drawing shows the dots as being at the mid-height of the numbers, not at the bottom as if they were decimals. In your instructions you represent the dots where you find them if they were used as decimals. Further, in your solution you use the dots to represent repeating digits. When I went to school we were taught to use a horizontal bar for this purpose. If you are going to use non-standard, or obsolete, rules for notation, it would be nice if you specified what they were.
Also a statement that only addition is allowed would not be out of order.
I presumed, incorrectly that the dots in the drawing were to be replaced by symbols for mathematical operations. With the addition of parentheses it was not difficult to come up with a (useless) solution.
The ellipsis in mathematics as I understand it means and so forth.
Half way down the page.
Normally dots should be used only where the pattern to be followed is clear, the exception being to show the indefinite continuation of an irrational number such as:
The last follows the definition given above.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.