Swammerdami
Squadron Leader
Despite the thread title I want to talk about a variant of the con-man's game Three-Card Monte.
Imagine a jar with two windows, one blue, one green. If you open the blue window you see EITHER a blue light OR an orangish-yellow light. If you open the green window you see EITHER a green light or an orangish-red light. But if you open the blue window the green window is disabled (and vice versa). You can perform EITHER the blue window experiment or the green window experiment, but not both.
The con-man(?) has three such jars; he re-initializes them in his special machine; he shuffles them up; and he presents them to you. There seem to be six bits of information involved (3 jars, each with 2 windows, each with 2 possible outcomes) but only 3 bits of information are accessible to you since opening one window disables the jar's other window.
You have nothing better to do than experiment for hours, letting the guy reset the jars, opening three windows of your choice and recording the results. You observe that if you open 1 green window and 2 blues (or 3 green windows and no blues), each of the eight possible results occurs about 12.5% of the time. BUT if you open 2 greens and a blue, or zero greens and 3 blues, four possible results become impossible!
Suppose now that you open 2 blue windows and see two orange lights. You know from Rule 2 that the 3rd blue window would reveal another orange light, so you don't waste your guess with that. You open the GREEN window on the 3rd jar. Let's say you see yet another orangish light.
You've used up your three experiments, but you can deduce what the other windows would have revealed had you opened them instead. We've already seen that Jar 3 would have shown orange were the Blue window opened. What about X?
Well, if you'd opened the three windows shown underlined in the table, Rule 1 dictates that X would show green. Similar reasoning dictates that Y would show green. So we can fill in the entire table:
So we've deduced all six bits! But suppose we had opened the three windows now underlined? What about Rule 1? Does the con-man have machines inside the jars communicating with each other by radio, and enforcing Rules 1 and 2 by flipping their programmed lights?
I'm already late for my afternoon errands, so I'll just leave these questions here for now.
Imagine a jar with two windows, one blue, one green. If you open the blue window you see EITHER a blue light OR an orangish-yellow light. If you open the green window you see EITHER a green light or an orangish-red light. But if you open the blue window the green window is disabled (and vice versa). You can perform EITHER the blue window experiment or the green window experiment, but not both.
The con-man(?) has three such jars; he re-initializes them in his special machine; he shuffles them up; and he presents them to you. There seem to be six bits of information involved (3 jars, each with 2 windows, each with 2 possible outcomes) but only 3 bits of information are accessible to you since opening one window disables the jar's other window.
You have nothing better to do than experiment for hours, letting the guy reset the jars, opening three windows of your choice and recording the results. You observe that if you open 1 green window and 2 blues (or 3 green windows and no blues), each of the eight possible results occurs about 12.5% of the time. BUT if you open 2 greens and a blue, or zero greens and 3 blues, four possible results become impossible!
- If you open 2 green windows and a blue window, you will always see an EVEN number of orangish lights.
- If you open zero greens and 3 blue windows, you will always see an ODD number of orangish lights.
Suppose now that you open 2 blue windows and see two orange lights. You know from Rule 2 that the 3rd blue window would reveal another orange light, so you don't waste your guess with that. You open the GREEN window on the 3rd jar. Let's say you see yet another orangish light.
- | Jar 1 | Jar 2 | Jar 3 |
---|---|---|---|
Blue Window | ORANGE | ORANGE | (orange) |
Green Window | X? | Y? | ORANGE |
Well, if you'd opened the three windows shown underlined in the table, Rule 1 dictates that X would show green. Similar reasoning dictates that Y would show green. So we can fill in the entire table:
- | Jar 1 | Jar 2 | Jar 3 |
---|---|---|---|
Blue Window | ORANGE | ORANGE | (orange) |
Green Window | (green) | (green) | ORANGE |
So we've deduced all six bits! But suppose we had opened the three windows now underlined? What about Rule 1? Does the con-man have machines inside the jars communicating with each other by radio, and enforcing Rules 1 and 2 by flipping their programmed lights?
I'm already late for my afternoon errands, so I'll just leave these questions here for now.