This post describes my solution to Matt Parker’s puzzle #8:
- The think-maths foldingchallenge.
- Matt Parker’s youtube channel.
My solution involves looking at the pattern of folds when viewing the folded stack from the sides. And then using some python code to generate all possible ways the folds can match up in a way that you will still end up with two rows of 4 sheets.
The list below shows all ways to fold a piece of paper, including ways where the top-left ends up on the inside of the folded stack.
Here I made some ascii drawings for the examples Matt showed in his video:
---------
A D E H
B C F G
---------
view from bottom view from left
-A----\ -A--\
-B--\ | -B--/
/----C--/ | -C--\
| /--D----/ -D--/
| \--E----\ -E--\
\----F--\ | -F--/
-G--/ | -G--\
-H----/ -H--/
---------
A H G D
B C F E
---------
view from bottom view from left
-A------\ -A--\
-B--\ | -B--/
/--C--/ | -C----\
| -D----\ | -D--\ |
| -E--\ | | -E--/ |
\--F--/ | | -F--\ |
/--G----/ | -G--/ |
\--H------/ -H----/
---------
A H B G
D E C F
---------
view from bottom view from left
-A--------\ -A----\
/----B------\ | -B--\ |
| /--C----\ | | -C--/ |
| | -D--\ | | | -D----/
| \--E--/ | | | -E----\
| -F----/ | | -F--\ |
| -G------/ | -G--/ |
\----H--------/ -H----/
This is a list of all ways to pair up sheets, and still have a valid folding.
40 with the ‘A’ on one of the outer corners of the sheet.
(AD, BC, EH, FG) (CF, DE) --> (AB, CD, EF, GH) (A, D, E, H) (B, C, F, G) << #0
(AH, BC, DG, EF) (CF, GH) --> (AB, CH, DE, FG) (E, F, C, B) (D, G, H, A) << #1
(AH, BG, CF, DE) (BH, CE) --> (AD, BC, EH, FG) (D, E, C, F) (A, H, B, G) << #2
(AB, CH, DG, EF) (BE, CD) --> (AH, BC, DE, FG) (A, B, E, F) (H, C, D, G)
(AB, CH, DE, FG) (BG, CD) --> (AH, BC, DG, EF) (A, B, G, F) (H, C, D, E)
(AB, CH, DE, FG) (BF, CE) --> (AH, BC, DG, EF) (A, B, F, G) (H, C, E, D)
(AB, CD, EH, FG) (BF, CE) --> (AD, BC, EF, GH) (A, B, F, G) (D, C, E, H)
(AB, CD, EF, GH) (BH, CE) --> (AD, BC, EH, FG) (D, C, E, F) (A, B, H, G)
(AB, CH, DG, EF) (BG, CF) --> (AH, BC, DE, FG) (E, F, C, H) (D, G, B, A)
(AF, BE, CD, GH) (BG, CF) --> (AH, BC, DE, FG) (A, F, C, D) (H, G, B, E)
(AB, CD, EF, GH) (BG, CF) --> (AD, BC, EH, FG) (D, C, F, E) (A, B, G, H)
(AH, BE, CD, FG) (BH, CG) --> (AF, BC, DE, GH) (D, C, G, F) (E, B, H, A)
(AB, CD, EH, FG) (BH, CG) --> (AD, BC, EF, GH) (D, C, G, F) (A, B, H, E)
(AB, CH, DG, EF) (BC, DE) --> (AF, BE, CD, GH) (F, E, D, G) (A, B, C, H)
(AB, CF, DE, GH) (BC, DG) --> (AH, BG, CD, EF) (H, G, D, E) (A, B, C, F)
(AB, CD, EF, GH) (BC, FG) --> (AH, BG, CF, DE) (H, G, F, E) (A, B, C, D)
(AF, BC, DE, GH) (BG, EF) --> (AH, BE, CD, FG) (A, F, E, D) (H, G, B, C)
(AB, CH, DG, EF) (BH, EG) --> (AF, BE, CD, GH) (D, G, E, F) (C, H, B, A)
(AH, BC, DE, FG) (BH, EG) --> (AF, BE, CD, GH) (D, E, G, F) (C, B, H, A)
(AB, CD, EF, GH) (BD, EG) --> (AH, BG, CF, DE) (C, D, B, A) (F, E, G, H)
(AD, BC, EH, FG) (BD, EG) --> (AH, BG, CF, DE) (A, D, B, C) (H, E, G, F)
(AB, CF, DE, GH) (BE, FG) --> (AH, BG, CD, EF) (D, E, B, A) (C, F, G, H)
(AD, BC, EF, GH) (CH, DE) --> (AB, CD, EH, FG) (B, C, H, G) (A, D, E, F)
(AF, BC, DE, GH) (CG, DF) --> (AH, BE, CD, FG) (A, F, D, E) (H, G, C, B)
(AD, BC, EF, GH) (CG, DF) --> (AB, CD, EH, FG) (A, D, F, E) (B, C, G, H)
(AH, BC, DE, FG) (CH, DG) --> (AF, BE, CD, GH) (E, D, G, F) (B, C, H, A)
(AD, BC, EH, FG) (CH, DG) --> (AB, CD, EF, GH) (A, D, G, F) (B, C, H, E)
(AH, BG, CF, DE) (CH, DG) --> (AB, CD, EF, GH) (B, G, D, E) (A, H, C, F)
(AD, BC, EH, FG) (CD, EF) --> (AH, BG, CF, DE) (G, F, E, H) (B, C, D, A)
(AH, BC, DG, EF) (CD, EH) --> (AB, CH, DE, FG) (A, H, E, F) (B, C, D, G)
(AH, BC, DE, FG) (CD, GH) --> (AB, CH, DG, EF) (A, H, G, F) (B, C, D, E)
(AH, BG, CD, EF) (CH, FG) --> (AB, CF, DE, GH) (A, H, C, D) (B, G, F, E)
(AF, BE, CD, GH) (CE, FH) --> (AB, CH, DG, EF) (G, H, F, A) (D, C, E, B)
(AH, BC, DE, FG) (CE, FH) --> (AB, CH, DG, EF) (A, H, F, G) (B, C, E, D)
(AF, BE, CD, GH) (DG, EF) --> (AH, BC, DE, FG) (C, D, G, H) (B, E, F, A)
(AH, BE, CD, FG) (DH, EG) --> (AF, BC, DE, GH) (B, E, G, F) (C, D, H, A)
(AH, BG, CD, EF) (DH, EG) --> (AB, CF, DE, GH) (B, G, E, F) (A, H, D, C)
(AF, BE, CD, GH) (DE, FG) --> (AB, CH, DG, EF) (B, E, D, C) (A, F, G, H)
(AH, BG, CF, DE) (EH, FG) --> (AB, CD, EF, GH) (A, H, E, D) (B, G, F, C)
(AH, BG, CF, DE) (EF, GH) --> (AD, BC, EH, FG) (C, F, E, D) (B, G, H, A)
40 with the ‘A’ somewhere in the center squares.
(AH, BG, CF, DE) (AD, BC) --> (AB, CD, EF, GH) (E, D, A, H) (F, C, B, G)
(AH, BG, CD, EF) (AF, BC) --> (AB, CF, DE, GH) (E, F, A, H) (D, C, B, G)
(AF, BE, CD, GH) (AH, BC) --> (AB, CH, DG, EF) (D, C, B, E) (G, H, A, F)
(AH, BC, DG, EF) (AE, BD) --> (AB, CH, DE, FG) (F, E, A, H) (G, D, B, C)
(AH, BG, CD, EF) (AE, BD) --> (AB, CF, DE, GH) (F, E, A, H) (C, D, B, G)
(AF, BE, CD, GH) (AG, BD) --> (AB, CH, DG, EF) (F, A, G, H) (E, B, D, C)
(AH, BC, DE, FG) (AG, BD) --> (AB, CH, DG, EF) (F, G, A, H) (E, D, B, C)
(AH, BC, DE, FG) (AF, BE) --> (AB, CH, DG, EF) (G, F, A, H) (D, E, B, C)
(AD, BC, EH, FG) (AF, BE) --> (AB, CD, EF, GH) (D, A, F, G) (C, B, E, H)
(AH, BG, CF, DE) (AF, BE) --> (AB, CD, EF, GH) (C, F, A, H) (D, E, B, G)
(AD, BC, EF, GH) (AH, BE) --> (AB, CD, EH, FG) (C, B, E, F) (D, A, H, G)
(AH, BC, DG, EF) (AG, BF) --> (AB, CH, DE, FG) (C, B, F, E) (H, A, G, D)
(AD, BC, EF, GH) (AG, BF) --> (AB, CD, EH, FG) (C, B, F, E) (D, A, G, H)
(AD, BC, EH, FG) (AH, BG) --> (AB, CD, EF, GH) (C, B, G, F) (D, A, H, E)
(AH, BG, CF, DE) (AB, CD) --> (AD, BC, EH, FG) (E, D, C, F) (H, A, B, G)
(AH, BE, CD, FG) (AB, CF) --> (AF, BC, DE, GH) (G, F, C, D) (H, A, B, E)
(AF, BE, CD, GH) (AB, CH) --> (AH, BC, DE, FG) (G, H, C, D) (F, A, B, E)
(AH, BC, DE, FG) (AB, EF) --> (AF, BE, CD, GH) (G, F, E, D) (H, A, B, C)
(AF, BC, DE, GH) (AB, EH) --> (AH, BE, CD, FG) (G, H, E, D) (F, A, B, C)
(AD, BC, EH, FG) (AB, GH) --> (AH, BG, CF, DE) (E, H, G, F) (D, A, B, C)
(AB, CD, EH, FG) (AF, DE) --> (AD, BC, EF, GH) (B, A, F, G) (C, D, E, H)
(AB, CD, EF, GH) (AH, DE) --> (AD, BC, EH, FG) (C, D, E, F) (B, A, H, G)
(AB, CD, EF, GH) (AG, DF) --> (AD, BC, EH, FG) (B, A, G, H) (C, D, F, E)
(AH, BG, CF, DE) (AG, DF) --> (AD, BC, EH, FG) (B, G, A, H) (C, F, D, E)
(AB, CD, EH, FG) (AH, DG) --> (AD, BC, EF, GH) (C, D, G, F) (B, A, H, E)
(AB, CH, DG, EF) (AC, DF) --> (AF, BE, CD, GH) (E, F, D, G) (B, A, C, H)
(AH, BC, DE, FG) (AC, DF) --> (AF, BE, CD, GH) (G, F, D, E) (H, A, C, B)
(AB, CF, DE, GH) (AC, DH) --> (AH, BG, CD, EF) (F, C, A, B) (E, D, H, G)
(AF, BC, DE, GH) (AC, DH) --> (AH, BE, CD, FG) (B, C, A, F) (E, D, H, G)
(AB, CD, EF, GH) (AC, FH) --> (AH, BG, CF, DE) (G, H, F, E) (B, A, C, D)
(AD, BC, EH, FG) (AC, FH) --> (AH, BG, CF, DE) (E, H, F, G) (D, A, C, B)
(AH, BE, CD, FG) (AD, EF) --> (AF, BC, DE, GH) (C, D, A, H) (B, E, F, G)
(AB, CD, EF, GH) (AD, EH) --> (AH, BG, CF, DE) (G, H, E, F) (B, A, D, C)
(AB, CH, DG, EF) (AD, EH) --> (AH, BC, DE, FG) (C, H, E, F) (B, A, D, G)
(AF, BE, CD, GH) (AD, EH) --> (AH, BC, DE, FG) (G, H, E, B) (F, A, D, C)
(AB, CH, DE, FG) (AD, GH) --> (AH, BC, DG, EF) (E, D, A, B) (F, G, H, C)
(AB, CH, DG, EF) (AH, FG) --> (AF, BE, CD, GH) (D, G, F, E) (C, H, A, B)
(AB, CF, DE, GH) (AE, FH) --> (AH, BG, CD, EF) (G, H, F, C) (B, A, E, D)
(AB, CH, DE, FG) (AE, FH) --> (AH, BC, DG, EF) (C, H, F, G) (B, A, E, D)
(AB, CH, DG, EF) (AF, GH) --> (AH, BC, DE, FG) (E, F, A, B) (D, G, H, C)
Here in a slightly different notation, using brackets to indicate which sheets are connected together.
40 with the ‘A’ on one of the outer corners of the sheet.
<(A(BC)D)(E(FG)H)> + <AB(C(DE)F)GH> -> <(AB)(CD)(EF)(GH)> << #0
<(A(BC)(D(EF)G)H)> + <AB(CDEF)(GH)> -> <(AB)(C(DE)(FG)H)> << #1
<(A(B(C(DE)F)G)H)> + <A(B(CDE)FGH)> -> <(A(BC)D)(E(FG)H)> << #2
<(AB)(C(D(EF)G)H)> + <A(B(CD)E)FGH> -> <(A(BC)(DE)(FG)H)>
<(AB)(C(DE)(FG)H)> + <A(B(CD)EFG)H> -> <(A(BC)(D(EF)G)H)>
<(AB)(C(DE)(FG)H)> + <A(B(CDE)F)GH> -> <(A(BC)(D(EF)G)H)>
<(AB)(CD)(E(FG)H)> + <A(B(CDE)F)GH> -> <(A(BC)D)(EF)(GH)>
<(AB)(CD)(EF)(GH)> + <A(B(CDE)FGH)> -> <(A(BC)D)(E(FG)H)>
<(AB)(C(D(EF)G)H)> + <A(B(CDEF)G)H> -> <(A(BC)(DE)(FG)H)>
<(A(B(CD)E)F)(GH)> + <A(B(CDEF)G)H> -> <(A(BC)(DE)(FG)H)>
<(AB)(CD)(EF)(GH)> + <A(B(CDEF)G)H> -> <(A(BC)D)(E(FG)H)>
<(A(B(CD)E)(FG)H)> + <A(B(CDEFG)H)> -> <(A(BC)(DE)F)(GH)>
<(AB)(CD)(E(FG)H)> + <A(B(CDEFG)H)> -> <(A(BC)D)(EF)(GH)>
<(AB)(C(D(EF)G)H)> + <A(BC)(DE)FGH> -> <(A(B(CD)E)F)(GH)>
<(AB)(C(DE)F)(GH)> + <A(BC)(DEFG)H> -> <(A(B(CD)(EF)G)H)>
<(AB)(CD)(EF)(GH)> + <A(BC)DE(FG)H> -> <(A(B(C(DE)F)G)H)>
<(A(BC)(DE)F)(GH)> + <A(BCD(EF)G)H> -> <(A(B(CD)E)(FG)H)>
<(AB)(C(D(EF)G)H)> + <A(BCD(EFG)H)> -> <(A(B(CD)E)F)(GH)>
<(A(BC)(DE)(FG)H)> + <A(BCD(EFG)H)> -> <(A(B(CD)E)F)(GH)>
<(AB)(CD)(EF)(GH)> + <A(BCD)(EFG)H> -> <(A(B(C(DE)F)G)H)>
<(A(BC)D)(E(FG)H)> + <A(BCD)(EFG)H> -> <(A(B(C(DE)F)G)H)>
<(AB)(C(DE)F)(GH)> + <A(BCDE)(FG)H> -> <(A(B(CD)(EF)G)H)>
<(A(BC)D)(EF)(GH)> + <AB(C(DE)FGH)> -> <(AB)(CD)(E(FG)H)>
<(A(BC)(DE)F)(GH)> + <AB(C(DEF)G)H> -> <(A(B(CD)E)(FG)H)>
<(A(BC)D)(EF)(GH)> + <AB(C(DEF)G)H> -> <(AB)(CD)(E(FG)H)>
<(A(BC)(DE)(FG)H)> + <AB(C(DEFG)H)> -> <(A(B(CD)E)F)(GH)>
<(A(BC)D)(E(FG)H)> + <AB(C(DEFG)H)> -> <(AB)(CD)(EF)(GH)>
<(A(B(C(DE)F)G)H)> + <AB(C(DEFG)H)> -> <(AB)(CD)(EF)(GH)>
<(A(BC)D)(E(FG)H)> + <AB(CD)(EF)GH> -> <(A(B(C(DE)F)G)H)>
<(A(BC)(D(EF)G)H)> + <AB(CD)(EFGH)> -> <(AB)(C(DE)(FG)H)>
<(A(BC)(DE)(FG)H)> + <AB(CD)EF(GH)> -> <(AB)(C(D(EF)G)H)>
<(A(B(CD)(EF)G)H)> + <AB(CDE(FG)H)> -> <(AB)(C(DE)F)(GH)>
<(A(B(CD)E)F)(GH)> + <AB(CDE)(FGH)> -> <(AB)(C(D(EF)G)H)>
<(A(BC)(DE)(FG)H)> + <AB(CDE)(FGH)> -> <(AB)(C(D(EF)G)H)>
<(A(B(CD)E)F)(GH)> + <ABC(D(EF)G)H> -> <(A(BC)(DE)(FG)H)>
<(A(B(CD)E)(FG)H)> + <ABC(D(EFG)H)> -> <(A(BC)(DE)F)(GH)>
<(A(B(CD)(EF)G)H)> + <ABC(D(EFG)H)> -> <(AB)(C(DE)F)(GH)>
<(A(B(CD)E)F)(GH)> + <ABC(DE)(FG)H> -> <(AB)(C(D(EF)G)H)>
<(A(B(C(DE)F)G)H)> + <ABCD(E(FG)H)> -> <(AB)(CD)(EF)(GH)>
<(A(B(C(DE)F)G)H)> + <ABCD(EF)(GH)> -> <(A(BC)D)(E(FG)H)>
40 with the ‘A’ somewhere in the center squares.
<(A(B(C(DE)F)G)H)> + <(A(BC)D)EFGH> -> <(AB)(CD)(EF)(GH)>
<(A(B(CD)(EF)G)H)> + <(A(BC)DEF)GH> -> <(AB)(C(DE)F)(GH)>
<(A(B(CD)E)F)(GH)> + <(A(BC)DEFGH)> -> <(AB)(C(D(EF)G)H)>
<(A(BC)(D(EF)G)H)> + <(A(BCD)E)FGH> -> <(AB)(C(DE)(FG)H)>
<(A(B(CD)(EF)G)H)> + <(A(BCD)E)FGH> -> <(AB)(C(DE)F)(GH)>
<(A(B(CD)E)F)(GH)> + <(A(BCD)EFG)H> -> <(AB)(C(D(EF)G)H)>
<(A(BC)(DE)(FG)H)> + <(A(BCD)EFG)H> -> <(AB)(C(D(EF)G)H)>
<(A(BC)(DE)(FG)H)> + <(A(BCDE)F)GH> -> <(AB)(C(D(EF)G)H)>
<(A(BC)D)(E(FG)H)> + <(A(BCDE)F)GH> -> <(AB)(CD)(EF)(GH)>
<(A(B(C(DE)F)G)H)> + <(A(BCDE)F)GH> -> <(AB)(CD)(EF)(GH)>
<(A(BC)D)(EF)(GH)> + <(A(BCDE)FGH)> -> <(AB)(CD)(E(FG)H)>
<(A(BC)(D(EF)G)H)> + <(A(BCDEF)G)H> -> <(AB)(C(DE)(FG)H)>
<(A(BC)D)(EF)(GH)> + <(A(BCDEF)G)H> -> <(AB)(CD)(E(FG)H)>
<(A(BC)D)(E(FG)H)> + <(A(BCDEFG)H)> -> <(AB)(CD)(EF)(GH)>
<(A(B(C(DE)F)G)H)> + <(AB)(CD)EFGH> -> <(A(BC)D)(E(FG)H)>
<(A(B(CD)E)(FG)H)> + <(AB)(CDEF)GH> -> <(A(BC)(DE)F)(GH)>
<(A(B(CD)E)F)(GH)> + <(AB)(CDEFGH)> -> <(A(BC)(DE)(FG)H)>
<(A(BC)(DE)(FG)H)> + <(AB)CD(EF)GH> -> <(A(B(CD)E)F)(GH)>
<(A(BC)(DE)F)(GH)> + <(AB)CD(EFGH)> -> <(A(B(CD)E)(FG)H)>
<(A(BC)D)(E(FG)H)> + <(AB)CDEF(GH)> -> <(A(B(C(DE)F)G)H)>
<(AB)(CD)(E(FG)H)> + <(ABC(DE)F)GH> -> <(A(BC)D)(EF)(GH)>
<(AB)(CD)(EF)(GH)> + <(ABC(DE)FGH)> -> <(A(BC)D)(E(FG)H)>
<(AB)(CD)(EF)(GH)> + <(ABC(DEF)G)H> -> <(A(BC)D)(E(FG)H)>
<(A(B(C(DE)F)G)H)> + <(ABC(DEF)G)H> -> <(A(BC)D)(E(FG)H)>
<(AB)(CD)(E(FG)H)> + <(ABC(DEFG)H)> -> <(A(BC)D)(EF)(GH)>
<(AB)(C(D(EF)G)H)> + <(ABC)(DEF)GH> -> <(A(B(CD)E)F)(GH)>
<(A(BC)(DE)(FG)H)> + <(ABC)(DEF)GH> -> <(A(B(CD)E)F)(GH)>
<(AB)(C(DE)F)(GH)> + <(ABC)(DEFGH)> -> <(A(B(CD)(EF)G)H)>
<(A(BC)(DE)F)(GH)> + <(ABC)(DEFGH)> -> <(A(B(CD)E)(FG)H)>
<(AB)(CD)(EF)(GH)> + <(ABC)DE(FGH)> -> <(A(B(C(DE)F)G)H)>
<(A(BC)D)(E(FG)H)> + <(ABC)DE(FGH)> -> <(A(B(C(DE)F)G)H)>
<(A(B(CD)E)(FG)H)> + <(ABCD)(EF)GH> -> <(A(BC)(DE)F)(GH)>
<(AB)(CD)(EF)(GH)> + <(ABCD)(EFGH)> -> <(A(B(C(DE)F)G)H)>
<(AB)(C(D(EF)G)H)> + <(ABCD)(EFGH)> -> <(A(BC)(DE)(FG)H)>
<(A(B(CD)E)F)(GH)> + <(ABCD)(EFGH)> -> <(A(BC)(DE)(FG)H)>
<(AB)(C(DE)(FG)H)> + <(ABCD)EF(GH)> -> <(A(BC)(D(EF)G)H)>
<(AB)(C(D(EF)G)H)> + <(ABCDE(FG)H)> -> <(A(B(CD)E)F)(GH)>
<(AB)(C(DE)F)(GH)> + <(ABCDE)(FGH)> -> <(A(B(CD)(EF)G)H)>
<(AB)(C(DE)(FG)H)> + <(ABCDE)(FGH)> -> <(A(BC)(D(EF)G)H)>
<(AB)(C(D(EF)G)H)> + <(ABCDEF)(GH)> -> <(A(BC)(DE)(FG)H)>
