It was stated in Tom and Ray’s answer that " the number of times The Counter has to TURN SWITCH A OFF is exactly …44 times…"
http://www.cartalk.com/content/prison-switcharoo-0?answer
NEVER IS IT NECESSARY FOR THE “COUNTER” TO TURN SWITCH “A” OFF 44 TIMES.
43 TIMES IF HE FINDS IT ON, 42 TIMES IF HE FINDS IT OFF HIS FIRST VISIT TO THE SWITCH ROOM.
PROOF BELOW:
NOTE: EACH NUMBER BELOW REPRESENTS A DIFFERENT PRISONER. THE ORDER DOESN’T MATTER; I PUT THEM IN NUMERICAL ORDER FOR ILLISTRATIVE AND SIMPLICITY SAKE, JUST TO REPRESENT 22 DIFFERENT PRISONERS TURNING SWITCH “A” ON BEFORE THE RETURN, EVENTUALLY, OF THE “COUNTER”. (1) MEANS A PRISONER MAY OR MAY NOT HAVE COME IN THE SWITCH ROOM AT THAT TIME. The following order leaves out any visit when switch A is not moved. (order of the 22 doesn’t matter, number of “leave switch A on” trips don’t matter because they tell the counter NOTHING ).
When the “counter” goes into the control room the first time there can be only two positions he can find switch “A”, ON or OFF. If he finds it “on” he knows either one prisoner has been there before him that has turned switch “A” “ON” OR it was left that way by the warden. If he finds it “off” he knows no prisoners have been there before him.
When THE “COUNTER” FINDS SWITCH “A” ON, THE “COUNTER” TURNS IT OFF EVERY TIME, EXCEPT FOR THE LAST TIME. (no need to on his last trip, he does not need any more info from any future prisoners after that).
(1)C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C
WHEN THE COUNTER FINDS SWITCH “A” ON FOR THE 44TH TIME, HE KNOWS 22 PRISONERS HAVE TURNED SWITCH “A” ON once AND HE KNOWS AT LEAST 21 prisoners HAVE TURNED SWITCH “A” ON A SECOND TIME. HE KNOWS ALL HAVE BEEN IN THE SWITCH ROOM AND HE HAS ONLY TURNED SWITCH “A” OFF 43 TIMES. THERE IS NO NEED TO TURN OFF SWITCH “A” THIS TRIP. IN FACT IF HE TURNS SWITCH “A” OFF A 44TH TIME AND WAITS FOR HIS RANDOM RETURN LOOKING FOR A 45TH FIND OF Switch “A” TO BE “ON”, HE WILL NEVER FIND IT, if the reason he first found it “on” was because a prisoner turned it on before his first visit, SO THEN he, THE “COUNTER” WILL REMAIN IN PRISON WITH NO EARLY RELEASE, EVER, BECAUSAE ALL PRISONERS HAVE ALREADY TURNED SWITCH “A” “ON” TWO TIMES AND HAVE FOLLOWED INSTRUCTIONS TO TURN “ON” SWITCH “A” 2 TIMES, NO MORE, NO LESS.
NOW, IF THE COUNTER COMES IN HIS FIRST TIME AND FINDS SWITCH “A” “OFF”, HE WILL KNOW NO OTHER PRISONER HAS BEEN THERE BEFORE HIM AND THE VISITS WOULD BE LIKE THIS:
C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C
AT HIS 43TH FINDING OF SWITCH “A” “ON”, THE COUNTER KNOWS ALL OTHER 22 PRISONERS HAVE TURNED SWITCH “A” ON ONCE. AND HE KNOWS ALL BUT ONE HAS TURNED SWITCH “A” ON A 2ND TIME, THOUGH THAT SECOND FACT REALLY DOESN’T MATTER IN THE LEAST TO HIM. ALL HE NEEDS TO KNOW IS THAT “ALL HAVE BEEN IN THE SWITCH ROOM, AT LEAST ONCE.” HE COULDN’T CARE LESS IF THE 22ND PRISONER EVER SHOWS UP FOR HIS SECOND TIME. IT DOESN’T MATTER. IF THE COUNTER WAITS FOR A 44TH CONFERMATION IT WOULD ONLY PROLONG HIS OWN STAY IN PRISON FOR WHO KNOWS HOW MUCH LONGER, CONSIDERING HOW LONG AT RANDOM IT WOULD TAKE THE WARDER TO PICK THAT 22ND PERSON FOR HIS 2ND TIME TO TURN SWITCH “A” “ON” AND AFTER THAT, RANDOMLY PICK THE “COUNTER” AGAIN.
WHEN THE COUNTER COMES IN AND FINDS SWITCH “A” HAS BEEN TURNED ON THE 43RD TIME, THERE IS NO NEED TO TURN SWITCH “A” OFF AGAIN. HE DOESN’T NEED ANY MORE INFORMATION TO KNOW “ALL PRISONERS HAVE BEEN IN THE SWITCH ROOM AT LEAST ONCE. HOW MANY NECESSARY SWITCH “OFFs”? 42 TIMES. REMEMBER ON HIS FIRST TRIP HE FOUND SWITCH “A” OFF ALLREADY, SO HE DOES NOT NEED TO TURN IT OFF THIS TIME. NEVER IS IT NECESSARY FOR THE COUNTER TO TURN SWITCH “A” OFF 44 TIMES, AS WAS STATED IN THE CAR TALK ANSWER.
I m sure you are right, but the use of capitols was so off putting that I did not get past the first couple paragraphs
If switch A is [ON] initially, the counter must find switch A [ON] 44 times to guarantee all other prisoners have been there at least once. It is true that he doesn’t have to turn it [OFF] the 44th time. But whether he turns it [OFF] or not that 44th time makes no difference in the outcome.