111 (x,y,z) position

11 is a perfect totient number.^[1]

111 is R₃ or the second repunit, a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are of the form 3n × 37. As a repunit, it also follows that 111 is a palindromic number.

All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 111₂ = 7₁₀. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 (sequence A002384 in the OEIS)

In base 18, the number 111 is 7³ (= 343₁₀) which is the only base where 111 is a perfect power.

The smallest magic square using only 1 and prime numbers has a magic constant of 111:

31	73	7
13	37	61
67	1	43

A six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:

1	11	31	29	19	20
2	22	24	25	8	30
3	33	26	23	17	9
34	27	10	12	21	7
35	14	15	16	18	13
36	4	5	6	28	32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

111 is also the magic constant of the n-Queens Problem for n = 6.^[2] It is also an nonagonal number.^[3]

In base 10, it is a Harshad number.^[4]