Number of ways to fold a 2x4 map

Question

M.Gardner in his book "Mathematical puzzles and diversions" states that the following 2x4 map

$$ \begin{array}{|c|c|} \hline 1 & \phantom{1} & \phantom{1} & \phantom{1} \\ \hline \phantom{1} & \phantom{1} & \phantom{1} & \phantom{1} \\ \hline \end{array} $$

can be folded in 40 different ways (along 7 segments shown) that the cell with 1 on it will be the upper one.

How to count all these ways and prove that there are exactly 40 of them?

Answer 1

For short hand: the horizontal line is $H$, the center vertical is $C$, the one to the left of $C$ is $L$, and the one to the right of $C$ is $R$. The cells are:

\begin{array}{|c|c|} \hline 1 & 2 & 3 & 4 \\ \hline 8 & 7 & 6 & 5 \\ \hline \end{array}

Fold $H$ last: 4 ways

$H$ and $L$ have 1 way to fold, so here can be ignored. As $C$ and $R$ have 2 ways to fold and their order doesn't matter (make dulplicate orientations) there are only $2\times2=4$ different ways to fold the sheet ignoring $H$ for the end.

Fold $H$ first: 7 ways

If one were to fold $H$ first, there are 6 different ways to fold the sheet. 4 are simply folding $R$ and $C$ in their 2 possible ways. The other 2 involve tucking cells $3-6$ between $1$ and $8$.

The seventh was well decribed by klm123 and aschepler: Fold $Ho$ . Slide $4-5$ between $1,8$. Making it more like a cylinder than a triangle, continue sliding in until $4-5$ are between $2,7$ and $3,6$ are between $1,8$. Then you can fully flatten the LoCoRo creases, making it a 1x1 square.

Fold $R$ then $H$ first: 6 ways

If one folds $R$ then $H$ one gets 6 orientations in exactly the same way as the first 6 option from folding $H$ first.

Fold $R$ and $C$ then $H$ first: + 4 ways

Neither $H$ last nor $L$ last scenereos can cover square $1$ as long as $H$ and $L$ are both folded out and no tucking possible as the first two folds quickly shorten the paper. This means that there are 8 possibilitys: $RoCoHoLo,RiCoHoLo,RoCiHoLo,RiCiHoLo$ and the same ones with $LoHo$ (but these 4 are duplicates of $H$ last).

Fold $R$ and $L$ then $H$ first: see below (Fold $L$ and $R$ then $H$ then $C$)

Fold $C$ then $H$ first: 3 ways

$C$ can now only be folded "out" in the same direction as $H$ or cell $1$ is covered.
Once this is done, however, $R$ can be folded into three different orientations. The options are to place $4$ and $5$ between $1$ and $8$,$2$ and $3$ or $7$ and $6$. Folding $L$ makes it a 1x1 but doesn't change the number of patterns.

Fold $C$ and $L$ then $H$ first: 3 ways

$C$ can now only be folded "in" and $L$ must be folded "out".
Similar to the previous case, there are 3 different options based on folding $R$ after $H$. They are to place to place $4$ and $5$ at the bottom, between $6$ and $7$ or between $2$ and $3$.

Fold $C$ and $R$ then $H$ first: see above (Fold $R$ and $C$ then $H$ first)

Fold $L$ then $H$ first: 7 ways

$L$ can now only be folded "out" and $H$ must be folded "out".
This yields 7 different possibilities based on where $3-6$ are placed. There is: one with $4-5$ between $2$ and $7$; 2 ($Ri$ and $Ro$) with $3-6$ between $1$ and $2$; 2 with $3-6$ between $7$ and $8$; and 2 with $3-6$ on the bottom.

Fold $L$ and $R$ then $H$ then $C$: 6 ways

$L$ can now only be folded "out".
This yields 6 different possibilities based on where $3-6$ are placed. There are: 2 ($Ri$ and $Ro$) with $3-6$ between $1$ and $2$; 2 with $3-6$ between $7$ and $8$; and 2 with $3-6$ on the bottom.

Answer 2

This is not the answer, but just one step forward it.
I give here 29 ways to fold the map + idea for 4 more ways.

I number the cells according to positions they would take at the resulting 1x1 folder map, counting from top to bottom.

Notations: H - fold along horizontal line; VI - fold along I-th vertical line; in - fold in a way that front part (usually with numbers) will be hidden inside; out - fold in a way that front part (usually with numbers) will be outside; "IJ" - cell in I-th row and J-th column.

Left-to-right order of vertical folds.

6 when fold horizontal first:

1. Hout, V1out, V2in, V3out $$ \begin{array}{|c|c|} \hline 1 & 4 & 5 & 7 \\ \hline 2 & 3 & 6 & 8 \\ \hline \end{array} $$
2. Hout, Vout, Vin, Vin $$ \begin{array}{|c|c|} \hline 1 & 4 & 7 & 6 \\ \hline 2 & 3 & 8 & 5 \\ \hline \end{array} $$
3. Hout, V1out, V2out, V3in $$ \begin{array}{|c|c|} \hline 1 & 8 & 5 & 4 \\ \hline 2 & 7 & 6 & 3 \\ \hline \end{array} $$
4. Hout, V1out, V2out, V3out $$ \begin{array}{|c|c|} \hline 1 & 8 & 3 & 6 \\ \hline 2 & 7 & 4 & 5 \\ \hline \end{array} $$
5. Hout, V1out, V2out, V3out (right end should go inside of left end) $$ \begin{array}{|c|c|} \hline 1 & 8 & 2 & 7 \\ \hline 4 & 5 & 3 & 6 \\ \hline \end{array} $$
6. Hout, V1out, V2out, V3out (folded half right end should go in-front of "21") $$ \begin{array}{|c|c|} \hline 1 & 8 & 2 & 5 \\ \hline 6 & 7 & 3 & 4 \\ \hline \end{array} $$

6 when fold horizontal 2-nd:

1. V1out, Hout, V2in, V3out (folded half must be put behind "1") $$ \begin{array}{|c|c|} \hline 1 & 6 & 5 & 2 \\ \hline 8 & 7 & 4 & 3 \\ \hline \end{array} $$
2. V1out, Hout, V2in, V3in (folded half must be put behind "11") $$ \begin{array}{|c|c|} \hline 1 & 6 & 3 & 4 \\ \hline 8 & 7 & 2 & 5 \\ \hline \end{array} $$
3. V1out, Hout, V2out, V3in $$ \begin{array}{|c|c|} \hline 1 & 2 & 6 & 7 \\ \hline 4 & 3 & 5 & 8 \\ \hline \end{array} $$
4. V1out, Hout, V2out, V3in (folded half must be put in-front of "21") $$ \begin{array}{|c|c|} \hline 1 & 2 & 5 & 6 \\ \hline 8 & 3 & 4 & 7 \\ \hline \end{array} $$
5. V1out, Hout, V2out, V3out $$ \begin{array}{|c|c|} \hline 1 & 2 & 8 & 5 \\ \hline 4 & 3 & 7 & 6 \\ \hline \end{array} $$
6. V1out, Hout, V2out, V3out (folded half must be put in-front of "21") $$ \begin{array}{|c|c|} \hline 1 & 2 & 7 & 4 \\ \hline 7 & 3 & 6 & 5 \\ \hline \end{array} $$

4 when fold horizontal 3-rd:

1. V1out, V2in, Hout, V3out $$ \begin{array}{|c|c|} \hline 1 & 6 & 3 & 8 \\ \hline 8 & 5 & 4 & 7 \\ \hline \end{array} $$
2. V1out, V2in, Hout, V3out (folded half must be put in-front of "21" and "22") $$ \begin{array}{|c|c|} \hline 1 & 6 & 3 & 8 \\ \hline 8 & 5 & 4 & 7 \\ \hline \end{array} $$
3. V1out, V2in, Hout, V3in (folded half must be put behind of "11" and "12") $$ \begin{array}{|c|c|} \hline 1 & 2 & 5 & 4 \\ \hline 8 & 7 & 6 & 3 \\ \hline \end{array} $$
4. V1out, V2out, Hout, V3in (impossible)
5. V1out, V2out, Hout, V3out (impossible)

4 when fold horizontal 4-th:

1. V1out, V2in, V3out, Hout $$ \begin{array}{|c|c|} \hline 1 & 2 & 3 & 4 \\ \hline 8 & 7 & 6 & 5 \\ \hline \end{array} $$
2. Vout, Vin, Vin, Hout $$ \begin{array}{|c|c|} \hline 1 & 2 & 4 & 3 \\ \hline 8 & 7 & 5 & 6 \\ \hline \end{array} $$
3. V1out, V2out, V3in, Hout $$ \begin{array}{|c|c|} \hline 1 & 4 & 3 & 2 \\ \hline 8 & 5 & 6 & 7 \\ \hline \end{array} $$
4. V1out, V2out, V3out, Hout $$ \begin{array}{|c|c|} \hline 1 & 4 & 2 & 3 \\ \hline 8 & 5 & 7 & 6 \\ \hline \end{array} $$

V2 first.

3 when fold horizontal 2-nd:

1. V2out, Hout, V1out, V3out $$ \begin{array}{|c|c|} \hline 1 & 8 & 7 & 2 \\ \hline 4 & 5 & 6 & 3 \\ \hline \end{array} $$
2. V2out, Hout, V1out, V3out ("14" goes behind "22") $$ \begin{array}{|c|c|} \hline 1 & 8 & 7 & 4 \\ \hline 2 & 3 & 6 & 5 \\ \hline \end{array} $$
3. V2out, Hout, V1out, V3in ("14" goes behind "13") $$ \begin{array}{|c|c|} \hline 1 & 8 & 5 & 7 \\ \hline 2 & 3 & 4 & 6 \\ \hline \end{array} $$

6 when fold horizontal 3-nd:

1. V2out, V1***, Hout (is impossible)
2. V2out, V3out, Hout, V1in $$ \begin{array}{|c|c|} \hline 1 & 8 & 6 & 7 \\ \hline 2 & 3 & 5 & 4 \\ \hline \end{array} $$

3. V2out, V3in, Hout, V1in $$ \begin{array}{|c|c|} \hline 1 & 8 & 7 & 6 \\ \hline 2 & 3 & 4 & 5 \\ \hline \end{array} $$

4. V2in, V1out, Hout, V1in ("14" in-front "13) $$ \begin{array}{|c|c|} \hline 1 & 8 & 5 & 4 \\ \hline 2 & 7 & 6 & 3 \\ \hline \end{array} $$

5. V2in, V1out, Hout, V1out ("14" behind "13) $$ \begin{array}{|c|c|} \hline 1 & 8 & 3 & 6 \\ \hline 2 & 7 & 4 & 5 \\ \hline \end{array} $$
6. V2out, V3out, Hout, V1in $$ \begin{array}{|c|c|} \hline 1 & 6 & 7 & 8 \\ \hline 2 & 5 & 4 & 3 \\ \hline \end{array} $$
7. V2out, V3in, Hout, V1in $$ \begin{array}{|c|c|} \hline 1 & 6 & 8 & 7 \\ \hline 2 & 5 & 3 & 4 \\ \hline \end{array} $$

V3 first. H 2-nd

1. V3in, Hout, V1out, V2out

2. V3in, Hout, V1out, V2in

3. V3out, Hout, V1out, V2out

4. V3out, Hout, V1out, V2in

(to be continued)