A380167 Maximum number of sets for the SET card game for n cards with 3 properties where each can take 3 values.

1, 1, 2, 3, 5, 8, 12, 12, 13, 14, 16, 19, 23, 26, 30, 36, 41, 47, 54, 62, 71, 81, 92, 104, 117

Offset

3,3

Comments

Also the maximum number of lines for n points in F_3^3 where a line is defined as three points p, q, r such that p+q+r = 0.

Links

Table of n, a(n) for n=3..27.

Justin Stevens and Duncan Wilson, The Maximum Number of Sets for 12 Cards is 14, arXiv:2501.12565 [math.CO], 2025.

Jim Vinci, The maximum number of sets for n cards and the total number of internal sets for all partitions of the deck, 2009.

Example

For n=3, the maximum number of SETs with 3 cards is 1 hence a(3)=1.
For n=4, no additional SETs can be formed, hence a(4)=1.
For n=5, we can take a 3-card SET, say {p, q, r}. Then, for two additional cards, s and t, outside of the SET, the only possible way this can form a SET is if we have one element from the first SET plus s and t form a SET. An example of this with the 4 properties being coordinates from 0 to 2 in mod 3 (in the equivalent definition of a SET) is the points {(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 0, 2), (0, 0, 1, 0), (0, 0, 2, 0)} which form 2 SETs. Hence a(5)=2.

Prog

(Python)
from itertools import combinations
def add_mod3(a, b, c):
    """Component-wise addition mod 3 of two 3D tuples."""
    return tuple((a[i] + b[i] + c[i]) % 3 for i in range(3))
def is_set(a, b, c):
    """Returns true if three elements form a set."""
    return add_mod3(a, b, c) == (0, 0, 0)
if __name__ == "__main__":
    all_points = [(x, y, z) for x in range(3) for y in range(3) for z in range(3)]
    max_sets = 0
    for n in range(3, 28):
        for comb in combinations(all_points, n):
            num_sets = 0
            for possible_set in combinations(comb, 3):
                if is_set(possible_set[0], possible_set[1], possible_set[2]):
                    num_sets += 1
            if num_sets > max_sets:
                max_sets = num_sets
        print("Max sets for %d cards: %d"%(n, max_sets))

Crossrefs

Cf. A090245 for a complementary sequence of the maximum number of cards with no sets.

Cf. A182240 for the number of ways to select n cards which is the search space complexity of this sequence for 4 properties instead of 3.

Sequence in context: A105766 A056695 A342746 * A287006 A020899 A057987

Adjacent sequences: A380164 A380165 A380166 * A380168 A380169 A380170

Keyword

nonn,fini,full

Author

Justin Stevens, Jan 22 2025

Status

approved