balance scale problem for 13 (not 12) items

Question

The 12-item balance scale puzzle is very familiar. The object is to find the lone non-standard item (if one exists) out of a group of 12 seemingly identical items, using a balance scale and a maximum of three weighings. Did you know that it is possible to accomplish the same outcome given a set of 13 seemingly identical items? I've scoured the web for a discussion of this problem/solution and have not found one. Am I the only one that has solved this problem?

BTW: If you allow four (4) weighings on a balance scale, how many seemingly identical items can you analyze and be assured of finding the lone non-standard item within the group?

If there is sufficient interest, I will publish the answers in a future post.

Answer 1

Solution for finding the odd marble from a set of thirteen (13) seemingly identical items

1ST WEIGHING - 5 ON 5, 2ND WEIGHING 3 ON 3, 3RD WEIGHING 1 ON 1. "S" REPRESENTS A MARBLE KNOWN TO BE OF STANDARD WEIGHT, BUT IS NOT ONE OF THE 13. USE OF THIS ITEM IS CRITICAL TO THE SOLVING THE PROBLEM

"/" => LEFT SIDE IS HEAVIER "\" => RIGHT SIDE IS HEAVIER "-" => BOTH SIDES ARE EQUAL

12345/6789S (OUTCOME 1A)

126/347

    1/2 => 1H
    1\2 => 2H
    1-2 => 7L

126\347

    3/4 => 3H
    3\4 => 4H
    3-4 => 6L

126-347

    8/9 => 9L
    8\9 => 8L
    8-9 => 5H

12345\6789S (OUTCOME 1B)

126\347

    1/2 => 2L
    1\2 => 1L
    1-2 =>7H

126/347

    3/4 => 4L
    3\4 => 3L
    3-4 => 6H

126-347

    8/9 => 8H
    8\9 => 9H
    8-9 => 5L

12345-6789S (OUTCOME 1C)

1011/12S (you can use "S" or any of 1 through 9)

    10/11 => 10H
    10\11 => 11H
    10-11 =>12L

1011\12S

    10/11 => 11L
    10\11 => 10L
    10-11 => 12H

1011 - 12S

    13/S => 13H
    13\S => 13L
    13-S => ALL IDENTICAL