Let z be the two-digit number you pick, x be the ten's digit of that number, y the one's digit, and c the final result of your calculation. Then your number is:
z = 10x + y
Follow the instrctions to subtract the sum of the digits from the original number z:
c = z - (x + y) = 10x + y - (x + y) = 9x
Look closely, and you will see that every multiple of nine on the board of symbols contains the same symbol. Hence, your "personal" symbol is known ahead of time, and set when they draw the board.
Didn't I ever tell you? Curmudgeons read minds too... Although, I guess if I could read minds, I would know whether or not I told you. Heck, even if didn't read minds, I ought to know! But what we curmudgeons truly are is forgetful...
a very old math trick.
there are quite a few that have to do with "9"...
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
In the land of Mordor, where the Shadows lie
-- JRRT
M.C. Escher - Reptilien