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In Heinlein's Space Cadet, one of many tests administered to the protagonist seems to have no way of scoring any points. See the excerpt below.

Late in the day he was ushered into a cubbyhole containing a chair, a gadget mounted on a desk, pencil and paper, and framed directions.
"If any score from a previous test," Matt read, "appears in the window marked SCORE, return the starting lever to the position marked NEUTRAL to clear the board for your test."
Matt found the window labeled "SCORE"; it had a score showing in it - "37."
Well, he thought, that gives me a mark to shoot at. He decided not to clear the board until he had read the instructions.

"After the test starts," he read, "a score of T will result each time you press the lefthand button except as otherwise provided here below. Press the lefthand button whenever the red light appears provided the green light is not lighted as well except that no button should be pressed when the righthand gate is open unless all lights are out. If the right-hand gate is open and the lefthand gate is closed, no score will result from pressing any button, but the lefthand button must nevertheless be pressed under these circumstances if all other conditions permit a button to be pressed before any score may be made in succeeding phases of the test. To put out the green light, press the righthand button. If the lefthand gate is not closed, no button may be pressed. If the lefthand gate is closed while the red light is lighted, do not press the lefthand button if the green light is out unless the righthand gate is open. To start the test move the starting lever from neutral all the way to the right. The test runs for two minutes from the time you move the starting lever to the right. Study these instructions, then select your own time for commencing the test. You are not permitted to ask questions of the examiner, so be sure that you understand the instructions. Make as high a score as possible."

"Whew!" said Matt.
Still, the test looked simple-one lever, two pushbuttons, two colored lights, two little gates. Once he mastered the instructions, it would be as easy as flying a kite, and a durn sight simpler than flying a copter! - Matt had had his copter license since he was twelve. He got to work.
First, he told himself, there seems to be just two ways to make a score, one with the red light on and one with both lights out and one gate open.
Now for the other instructions - Let's see, if the lefthand gate is not closed - no, if the lefthand gate is closed - he stopped and read them over again.
Some minutes later he had sixteen possible positions of gates and conditions of lights listed. He checked them against the instructions, seeking scoring combinations. When he was through he stared at the result, then checked everything over again.
After rechecking he stared at the paper, whistled tunelessly, and scratched his head. Then he picked up the paper, left the booth, and went to the examiner.
That official looked up. "No questions, please."
"I don't have a question," Matt said. "I want to report something. There's something wrong with that test. Maybe the wrong instructions sheet was put in there. In any case, there is no possible way to make a score under the instructions that are in there."

Was Matt's conclusion correct?

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It depends on the precise mechanics of the game, and the starting state.

I can't find any rule forbidding pushing buttons when both gates are closed, except when the red light is on and the green is off (which has contradictory rules), so if the game starts in one of those positions, he should be able to score. If the game does not start in one of those positions, and the board can only be changed by pressing a button, he's out of luck.

The table of specified moves comes out to this:

\begin{array}{l|l|l|l|l} \text{red light} & \text{green light} & \text{left gate} & \text{right gate} & \text{button} \\ \hline \text{on} & \text{on} & \text{open} & \text{open} & \text{None} \\ \text{on} & \text{on} & \text{open} & \text{closed} & \text{None} \\ \text{on} & \text{on} & \text{closed} & \text{open} & \text{None} \\ \text{on} & \text{on} & \text{closed} & \text{closed} & \\ \text{on} & \text{off} & \text{open} & \text{open} & \text{None} \\ \text{on} & \text{off} & \text{open} & \text{closed} & \text{None} \\ \text{on} & \text{off} & \text{closed} & \text{open} & \text{None} \\ \text{on} & \text{off} & \text{closed} & \text{closed} & *** \\ \text{off} & \text{on} & \text{open} & \text{open} & \text{None} \\ \text{off} & \text{on} & \text{open} & \text{closed} & \text{None} \\ \text{off} & \text{on} & \text{closed} & \text{open} & \text{None} \\ \text{off} & \text{on} & \text{closed} & \text{closed} & \\ \text{off} & \text{off} & \text{open} & \text{open} & \text{None} \\ \text{off} & \text{off} & \text{open} & \text{closed} & \text{None} \\ \text{off} & \text{off} & \text{closed} & \text{open} & [\text{No Score}] \\ \text{off} & \text{off} & \text{closed} & \text{closed} & \\ \end{array}

The asterisks (on/off/closed/closed) mark a contradictory row, with one direction specifying pressing the left button and the other prohibiting it.

Text of the "No Score" row:

the lefthand button must nevertheless be pressed under these circumstances if all other conditions permit a button to be pressed before any score may be made in succeeding phases of the test.

The rules don't seem to prohibit pushing a button, but he won't get a score from it.

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The rules do not explicitly forbid pressing the button when not ordered to do so by the rules. It is not possible to score at the times you are instructed to press the button, but it is possible to score at other times.

Here's the matrix I used to determine what's allowed.

            Left/Right  OO OC CO CC
Red/Green
       00               11 11 11 11
       01               11 11 11 11
       10               11 11 11 11
       11               11 11 11 11

The left digit of a value indicates whether the left button may be pressed in that state. The right digit is the right button.

I proceeded to eliminate my options like so:

If the lefthand gate is not closed, no button may be pressed.

00 00 11 11
00 00 11 11
00 00 11 11
00 00 11 11

No button should be pressed when the righthand gate is open unless all lights are out.

00 00 11 11
00 00 00 11
00 00 00 11
00 00 00 11

If the lefthand gate is closed while the red light is lighted, do not press the lefthand button if the green light is out unless the righthand gate is open.

00 00 11 11
00 00 00 11
00 00 00 01
00 00 00 11

At this point, this command doesn't apply:

Press the lefthand button whenever the red light appears provided the green light is not lighted as well.

Since pressing the right button and turning off the green light will never send you from a 0 to a 1 for the left button, it can be ignored.

If the right-hand gate is open and the lefthand gate is closed, no score will result from pressing any button, but the lefthand button must nevertheless be pressed under these circumstances if all other conditions permit a button to be pressed before any score may be made in succeeding phases of the test.

This command must be followed if both lights are off, like this:

00 00 10 00
00 00 00 00
00 00 00 00
00 00 00 00

Other than these possibilities, the only way you can score points by pressing the lefthand button are these 3 states:

00 00 00 10
00 00 00 10
00 00 00 00
00 00 00 10

If the rules specified that you cannot score points unless instructed to press the left button, then it would indeed be impossible.

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  • $\begingroup$ I read "Press the lefthand button whenever the red light appears provided the green light is not lighted as well." to mean you cannot press he lefthand button if both the red and green lights are lite." Neverthe less, if both gates are close and the green light is off, you can press the lefthand button to score. $\endgroup$ – kaine May 22 '14 at 18:05
  • $\begingroup$ @kaine I read it literally as saying you must press the button then, not as saying you may only press the button then. However, Matt probably interpreted it the same way as you did. It's somewhat ambiguous. $\endgroup$ – Kendall Frey May 22 '14 at 18:39
  • $\begingroup$ I didn't read it as you may only press the button then (as I agree Matt may have) but that you explicitly cannot press it if the green light is lighted as well (from the command you say doesn't apply). If it doesn't say not to press it though (as it does for the other two scoring cases) press it all you want. That is why I voted up your answer. $\endgroup$ – kaine May 22 '14 at 18:59
  • $\begingroup$ So this means that if the starting condition is that both gates are closed and either red light is off or green light is on, only then he can score? Or does the rule say that the both off, left-closed, right-open must happen first before you can score? $\endgroup$ – justhalf Jul 4 '14 at 2:46

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