Pentahex Pair Oddities


A polyhex oddity is a symmetrical figure formed by an odd number of copies of a polyhex. Symmetrical figures can also be formed with copies of two different polyhexes. Here are the smallest known full-symmetry oddities for the 231 pairs of polyhexes.

See also Pentomino Pair Oddities and Hexiamond Pair Oddities.

Helmut Postl improved on some of my solutions.

Carl and Johann Schwenke improved many of my solutions and solved cases that were previously unsolved.

Chronology of Solutions


Basic Solutions

5AC —5AD 115AE 175AF 175AH 17
5AI 235AJ 175AK 235AL 175AN 11
5AP 115AQ 175AR 175AS 295AT 53
5AU 295AV 235AW 175AX 295AY 17
5AZ 235CD 115CE 235CF 175CH 23
5CI 475CJ 295CK 175CL 17
5CN 175CP 115CQ 35
5CR 295CS 235CT —5CU 175CV 29
5CW 355CX 595CY 175CZ 175DE 11
5DF 115DH 115DI 115DJ 115DK 11
5DL 115DN 115DP 115DQ 115DR 11
5DS 115DT 115DU 115DV 115DW 11
5DX 115DY 115DZ 115EF 175EH 23
5EI 175EJ 235EK 175EL 175EN 17
5EP 115EQ 235ER 175ES 175ET —
5EU 295EV 295EW 295EX 175EY 17
5EZ 175FH 175FI 175FJ 115FK 17
5FL 175FN 175FP 115FQ 175FR 17
5FS 175FT 175FU 235FV 175FW 17
5FX 175FY 175FZ 175HI 55HJ 17
5HK 115HL 175HN 115HP 115HQ 17
5HR 175HS 115HT 175HU 235HV 17
5HW 235HX 115HY 115HZ 175IJ 17
5IK 235IL 115IN 115IP 115IQ 23
5IR 175IS 475IT 415IU 235IV 35
5IW 295IX 655IY 115IZ 175JK 17
5JL 175JN 115JP 115JQ 175JR 11
5JS 175JT 235JU 235JV 175JW 29
5JX 175JY 115JZ 175KL 175KN 11
5KP 115KQ 115KR 115KS 115KT 23
5KU 115KV 235KW 115KX 115KY 11
5KZ 115LN 115LP 115LQ 175LR 17
5LS 115LT 235LU 175LV 235LW 23
5LX 235LY 115LZ 115NP 115NQ 11
5NR 115NS 175NT 115NU 115NV 11
5NW 175NX 175NY 115NZ 175PQ 11
5PR 115PS 115PT 115PU 115PV 11
5PW 115PX 115PY 115PZ 115QR 17
5QS 295QT 235QU 115QV 235QW 5
5QX 235QY 175QZ 115RS 235RT 17
5RU 175RV 175RW 175RX 175RY 11
5RZ 175ST 295SU 235SV 235SW 23
5SX 535SY 115SZ 235TU 355TV 23
5TW 175TX 295TY 175TZ 235UV 23
5UW 175UX 115UY 175UZ 235VW 29
5VX 295VY 175VZ 175WX 115WY 17
5WZ 175XY 115XZ 115YZ 11

Lesser Symmetries

I know of no full oddities for the pairs A-C, C-T, and E-T. Here are the cell-centered oddities or tri-oddities for these pairs with the highest known symmetries.

A and C

C and T

E and T

Holeless Variants

5AE 235AF 295AI 535AS 535AT 59
5AU —5AV 295AX 535CE —5CF 41
5CH 295CI —5CJ 355CN 235CQ —
5CS 475CU 595CW 415CX 835EF 29
5EI 835EJ 295EK 295ES 655EU 47
5EV 415EX 535FH 235FI 595FJ 35
5FK 235FL 235FS 295FT 355FU 59
5FV 415FX 415HI 115HJ 295HL 23
5HT 295HV 295HW 295IK 295IN 23
5IS —5IT —5IU —5IW 415IX —
5IY 175JQ 295JT 475JU 475JV 23
5JW 415KT 295LU 235NT 295NU 23
5NV 175QR 235QS 355QU 235QW 17
5RS 295RT 235RU 235RY 175ST 83
5SU 415SV 295SX 595SY 175SZ 29
5TU 535TV 355TW 415TX 835UV 29
5UW 415UX —5VW 415XY 17

Last revised 2023-03-03.

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Col. George Sicherman [ HOME | MAIL ]