# Flat Hexacube Compatibility

## Introduction

A hexacube is a solid made of six cubes joined face to face. A solid hexomino, or flat hexacube, is a hexacube whose cells lie all in one plane. As there are 35 hexominoes, there are 35 flat hexacubes:

Two or more polyforms are compatible if there is a polyform that each can tile. Most pairs of hexominoes are compatible, but many are not; see Giovanni Resta's page Hexominoes. All but one pair of flat hexacubes are known to be compatible:

If you solve this case, or find a smaller solution for another case, please write.

Thanks to Mark Smith for suggesting this problem.

## Table of Tiles

This table shows how many tiles are needed to demonstrate compatibility of two flat hexacubes. Green cells indicate that the number is equal to that for the corresponding hexominoes. Purple cells indicate that no solution is known for the corresponding hexominoes.

1234567891011121314151617181920212223242526272829303132333435
12224222444442244463446346664626366
22222224222224422243222222222224448
32222244222222224426224622222442622
422224242222222222222222234422342108
54222423222222224443264422422226668
62224422424244484286262242242438622
72242222224222424223222244432222334
824443222422442424232424244424266?16
942222422224442422222242624422344148
104222222422222444466222422423238642
1142222442222242424264244422102842222
1242222222422484422432222822222243410
134222242442242222222224224222223222
142422242442482224422248283444232244
1524222442242422222262244484644236124
164222282444442224822248426282824488
174242444224222424224223342242444342
184242422424422428223222382222422424
196422482226242222222222244244242224
2033623633266322624322664644364322610
214222222222422222222222222422222424
224222662422222424222622224282644284
236242422242424848322622242423244422
243262422424422244332422242244242446
2542222442624828424846224422244246124
266223224422224386224424222224342222
276224424444222442222442422224444622
28622424344210224684243282422224326616
2942222222232224422246223444422264812
306242242422822248442426224344234648
312243232233422322424324442443234366
326424682648243234422224424242644242
333462663646232264342242446266463244
3464210623?14424241284226282412226846442
35682882416822102448244104426422161286242
1234567891011121314151617181920212223242526272829303132333435

## Solutions

There are 595 cases, too many to show here. Instead I show only solutions with at least 10 tiles. The flat solution is from Resta's page.

### 16 Tiles

Last revised 2015-11-25.

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Col. George Sicherman [ HOME | MAIL ]