What are these cards for?

March 14, 2022 12:56 PM

Here are some cards. I don't know what they are for. Maybe you know what they are for? One,Two, and Three
posted by Just this guy, y'know to Grab Bag (11 answers total)

(mathematician here)

I don't know where the cards come from, but those cards are definitely representing permutations. S_n represents the group of all permutations of n elements. There should be six total cards for S₃, for instance, representing all six ways of ordering a set of three elements.

The cards with A₄ on them are representing so-called alternating permutations.

Putting the cards next to each other represents what happens when you do one permutation followed by another.

Is there anything on the backs of the cards?
posted by number9dream at 1:07 PM on March 14, 2022

One thing that might help if you use Google Chrome: Take a picture of one playing card, on its own, with a clear, non-grain background. Open that image in a new tab. Right-click on that image and select "Search Image with Google Lens" to look for nearest matches.

I did the above on the three links you provided, but I didn't get results. It might be easier to find a match if you use an image with only one card. Hope this is useful in any way.
posted by They sucked his brains out! at 1:08 PM on March 14, 2022

Nothing further to add, but I'm very curious to know where these cards came from.
posted by number9dream at 2:12 PM on March 14, 2022

The card labeled "e" in set one connects every object to itself, and 'e' is a standard designation for the identity element of groups.
posted by jamjam at 2:42 PM on March 14, 2022

These are definitely “answer keys” for a learning toy like this. Basically the kid wraps the string from the question on the left side to the matching answer on the right side, and the teacher can check the solution by comparing the string pattern on the back to this card.
posted by mekily at 3:14 PM on March 14, 2022

Agree it's likely a many to many "match key" solution. You often see this in proficiency exams where you need to match a list of items in A to B.
posted by kschang at 3:28 PM on March 14, 2022

I want to reinforce that these are visual representations of group elements in mathematics, specifically abstract algebra. That's what the labels mean. I'm browsing in bed but I will take a closer look in the morning and see if I can be more specific.
posted by dbx at 7:58 PM on March 14, 2022

I want to reinforce that these are visual representations of group elements in mathematics, specifically abstract algebra.

Thank you!

Folks, this is not a guess. Look at picture #3, for instance. Take the card on the left in the middle row, with the label a at the top. If you imagine labeling the dots on each side from 1 to 8, top to bottom, and then look at which numbers the lines connect (reading from left to right), this card represents the following permutation:

• 1 maps to 2
• 2 maps to 3
• 3 maps to 4
• 4 maps to 1
• 5 maps to 6
• 6 maps to 7
• 7 maps to 8
• 8 maps to 5

In abstract algebra, the symbol a² means "do the transformation a twice". What happens if we do that? First 1 maps to 2, then 2 maps to 3, so that means a² maps 1 to 3. Similarly, first 2 maps to 3, then 3 maps to 4, so a² maps 2 to 4. Here's the full list for a²:

• 1 maps to 3
• 2 maps to 4
• 3 maps to 1
• 4 maps to 2
• 5 maps to 7
• 6 maps to 8
• 7 maps to 5
• 8 maps to 6

That is exactly the permutation that is represented by the card that is third from the right in the middle row, which is labeled a² at the top.

If you do the permutation a a third time, what happens? Now:

• 1 maps to 4
• 2 maps to 1
• 3 maps to 2
• 4 maps to 3
• 5 maps to 8
• 6 maps to 5
• 7 maps to 6
• 8 maps to 7

That is exactly the permutation represented by the second card from the right in the middle row, which is labeled a³.

These are permutations.
posted by number9dream at 7:20 AM on March 15, 2022

To add a little more:

Reiterating what jamjam said above, in abstract algebra, the symbol e is often used to denote the identity element of a group. That is, it represents a neutral transformation that maps each number to itself. That is exactly what you see in the rightmost card in the middle row of picture #3.

There is also the permutation labeled b, the central card in the middle row of that picture. If you do the transformation a three times, followed by transformation b, you get exactly the transformation represented by the card labeled a³b.

In the bottom row, there are five cards labeled S₃. These cards represent five of the six possible permutations of three elements. Too bad one is missing!

There are 24 permutations of a set of four elements, and exactly half of them are "even". These are the 12 permutations in the top row that are labeled as A₄. This is standard mathematical notation, as shown in the Wikipedia page on the alternating group that I linked to above.

OP, could you please say where you found these cards?
posted by number9dream at 7:41 AM on March 15, 2022

Thanks all. Based on some of the responses I did some research and determined that the Q cards are probably the Quaternion group , whilst the S and A cards are the permutations as mentioned above.

A friend asked me I could figure out what the cards were. They used to belong to their grandmother, who was a secondary school maths teacher.
posted by Just this guy, y'know at 8:59 AM on March 15, 2022

the Q cards are probably the Quaternion group, whilst the S and A cards are the permutations as mentioned above

Even better than just plain quaternions, this is a representation of the quaternion group Q as a subgroup of the permutation group S₄! (See Cayley's Theorem, which says that every "abstract" group can be represented as a subgroup of a permutation group.)

This is my favorite AskMefi in a long time!
posted by number9dream at 11:03 AM on March 15, 2022