Black Box: The Discovery Game
Black Box is a game designed to simulate the process of scientific discovery. The game is divided into a series of sessions. In each session, one of the players assumes the role of Nature; the others play the role of scientists. The role of Nature rotates among the players from session to session so that, ideally, every player gets a chance to be Nature at least once during the game. (For descriptive convenience, I will refer to Nature using the feminine pronoun, as in “Mother Nature,” and the scientists using the masculine pronoun, with the clear understanding that the gender refers to the role and not to the player occupying that role.)
At the beginning of each session, Nature devises a process. A process, as viewed here, is an operation that transforms an input into an output. An example of a process is addition. In this case, the input consists of two numbers, and the output is a single number, namely, the sum of those two numbers.
We require that all processes be deterministic; no random elements are allowed. Also, we require that the output be determined solely by the input and not by any exogenous factors such as the weather, time of day, or any characteristic of the person providing the input. Finally, we require that the output be unambiguously determined by the input. This latter requirement would rule out, for example, a process whose output is either “yes” or “no,” depending on the answer to the question “Is the input the name of a famous musician?”
After Nature has devised the process, she records a description of that process for future reference and verification. This description may be a written description, a mathematical formula that computes the output as a function of the input, or an algorithm implemented on a computer. Nature then announces to the scientists the format of the input. In the addition example that we are using, Nature may say something like “The input consists of two integers, each between 0 and 100.” (Nature may impose range restrictions on the input for computational convenience.)
It is the goal of the scientists to discover Nature’s process. To this end, they perform simulated experiments by providing Nature with inputs and observing the corresponding outputs produced by Nature.
For descriptive convenience, imagine that the players are seated in a circle, with play beginning with the scientist to Nature’s immediate left taking his turn, and proceeding clockwise around the circle until all of the scientists have had a turn, at which time the first round is concluded. Play for the session continues for 20 rounds or until all of the scientists have discovered the process, whichever comes first.
A scientist on his turn has three options. He may provide Nature with an input in the prescribed format. For the addition example that we are using, he may say something like “two and three.” Nature then performs the process, addition in this case, and then announces the output, namely, five. All inputs and outputs are available to all of the scientists, who may record this information for future reference. A display device such as a whiteboard or computer spreadsheet projected onto a screen may be used by Nature to display this and other pertinent information for the convenience of the other players.
If a scientist believes that he has discovered the process, he may on his turn say “I’d like to be tested,” or words to that effect. In this case, the scientist to the immediate left of the scientist to be tested provides an input. However, instead of Nature performing the process and producing the output, it is left to the testee to produce the output. If the testee produces the correct output as judged by Nature, then Nature will say “correct”; otherwise, Nature will say “No, the correct output is <correct output>.”
If the testee successfully passes that test, he is similarly tested by the next scientist in line to take his turn, and if he passes the second test, he is tested once more by the third scientist in line.
If the testee successfully passes all three tests, he is deemed to have discovered the process and is awarded points for his discovery, the number of points depending on the round in which he discovers the process. If he discovers the process in the first round, he receives 20 points, that number of points decreasing by one for each round that goes by. A scientist who hasn’t discovered the process by the end of the 20th round will, of course, receive no points for the session. A scientist who has discovered the process no longer participates in providing inputs or being tested, but remains in place in order to test the other scientists as the need arises. Whether or not the former testee has discovered the process, play resumes with the scientist to the immediate left of the former testee taking his turn.
If a scientist on his turn chooses to neither provide an input nor be tested, or if he has already discovered the process, he will say “pass” in order to alert the next scientist in line to take his turn. A scientist who passes is free to participate in subsequent rounds when it becomes his turn to do so (assuming, of course, that he hasn’t yet discovered the process).
Given players of comparable levels of skill, it is generally possible for any one of those players to devise a process that is so complex that it would be unlikely that the other players would be able to discover the process in the 20 rounds available. Thus, it would appear to be a good strategy for the player who is Nature to devise a process that is exceeding difficult to discover, thus rendering the other players scoreless for that session, and thereby gaining a relative advantage. If all players adopted this strategy, then all of the processes would be so difficult to discover that the game would be uninteresting.
Ideally, we would like Nature to devise a process that is challenging but not excessively difficult. In order to motivate her to do so, she is awarded points for the session, computed as follows. Calculate the average score of the scientists for the session. If that average score is equal to or less than 10, double that value and round off to the nearest integer to get Nature’s score for the session. If the average score is greater than 10, subtract it from 20, double the result, and round off to the nearest integer to get Nature’s score.
Regarding the condition for ending the game, there are two possibilities. One possibility is that the game ends after the score of any player equals or exceeds a predetermined value, such as 50 or 100 points. The other alternative is that the game continues until all of the players have had a chance to be Nature once or a predetermined number of times.
Regarding the number of players, the game requires a bare minimum of four players: one to be Nature and at least three scientists, so that every scientist can be tested by two other scientists. While there is no upper limit to the number of players, it should be noted that the time required for every player to be Nature once or a given number of times is approximately proportional to the square of the number of players.
One of the benefits of playing Black Box, other than the enjoyment of the game, is that it provides the players with an opportunity to practice and thereby improve their pattern recognition skills. Questions requiring pattern recognition ability, such as analogies and complete‑the‑series, appear prominently on intelligence tests, which are used as admission criteria for high IQ societies such as Mensa, and on similar tests such as the SAT, a high score on which increases your chances of gaining admission to the college of your choice.
As I write this, we are in the midst of the coronavirus pandemic, which requires physical distancing and in some cases isolation. Black Box does not require any physical interaction between the players, only the exchange of information, so it can be played remotely.