# Tiling rectangles with Hexomino plus rectangle #3

Previous puzzle in this series: Tiling rectangles with Hexomino plus rectangle #2

The goal is to tile rectangles as small as possible with the given hexomino, in this case number 3 of the 25 hexominoes which cannot tile a rectangle alone. We allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one of the given hexomino will tile.

Example with the $1\times 1$, $1\times 2$, $1\times 3$ or $2\times 3$ you can tile a $3\times 4$ as follows:

Now we don't need to consider $1\times 1$, $1\times 2$, $1\times 3$ or $2\times 3$ further as we have found the smallest rectangle tilable with copies of the hexomino plus copies of those rectangles.

This is split into two sections: Those tilable by hand, and those probably requiring a computer. Feel free to solve any of them by hand, but please don't post computer-found tilings for those in the no-computer section.

No Computer section All of these should be tiled by hand only. This also means please don't look up answers on the web... This does not preclude you from for example using an image program to manipulate shapes on the screen, just from using a computer to search for or automate the arrangement.

• Width 1: Lengths 1 to 3 (given), 4 to 8
• Width 2: Lengths 2, 3 (given), 4 to 9
• Width 3: Lengths 4 5 8
• Width 4: Lengths 4 (difficult but worth it) 5 6
• Width 5: Lengths 5 6

Computer section Master solvers may well solve these by hand.

• Width 1: Lengths 9 to 12
• Width 2: Lengths 10 12 14
• Width 3: Length 7
• Width 4: Length 7

Here are attempts for 2x2:

4x4=16

For 3x4:

4x6=24

This one for 1x4 is slightly smaller:

4x5=20

for 2x6 (and 1x6):

3x8=24

and for 4x6

6x8=48

and for 2x4

4x8=32

Here's one for 2x7 (which works for 1x7 as well):

5x8=40

and here is its 'inverse', for 3x5 and 1x5:

7x6=42

I found a computer-generated one in my archives, for 1x9:

9x23=207

• 2x2=4x4 is minimal, 3x4=4x6 is minimal, 1x4=4x6 is not, 1x6,2x6=3x8 both minimal, 4x6=6x8 is, 2x4=6x8 is not Jun 14 '18 at 5:46
• Here are some updates. Solving these while driving to the office is hard ... Jun 14 '18 at 6:34
• 1x4=4x5, 2x4=4x8 both minimal Jun 14 '18 at 6:56

Minimal 5x5:

7 x 7 = 49

Minimal 1x8:

6 x 10 = 60