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 Numerical path (Posted on 2008-09-23)
Draw only one continuous line into the 8x8 diagram below, which begins in the left cell above (with a “01”) and ends in the right bottom cell (with a “33”). The line must pass through all the numbers from 01 to 33 (through the center of each cell), once and only once through each one, and can run only horizontally or vertically (not diagonally). Besides, the line may not cross itself.

Far from being a trial and error problem, this can be solved logically.
```     +-----+-----+-----+-----+-----+-----+-----+-----+
| 01  | 16  | 05  | 20  | 25  | 09  | 21  | 01  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 18  | 10  | 27  | 26  | 11  | 17  | 12  | 32  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 32  | 11  | 15  | 19  | 08  | 06  | 27  | 20  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 17  | 04  | 13  | 24  | 30  | 28  | 31  | 02  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 25  | 10  | 02  | 26  | 04  | 28  | 22  | 13  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 05  | 14  | 30  | 08  | 15  | 31  | 19  | 06  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 23  | 07  | 24  | 16  | 29  | 22  | 18  | 19  |
+-----+-----+-----+-----+-----+-----+-----+-----+
| 03  | 12  | 09  | 03  | 07  | 14  | 23  | 33  |
+-----+-----+-----+-----+-----+-----+-----+-----+```

 See The Solution Submitted by pcbouhid Rating: 2.5000 (2 votes)

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 Analytical solutions (hopefully all of them) | Comment 7 of 11 |
First I looked through the numbers for any numbers that only appeared once. 21 and 29 were the only ones. (Every other number but 19 only appeared twice.) Then I crossed off the 1 in the upper right (since 1 was the beginning number), and this meant 9 and 12 had to be in the path. So I crossed those off on the bottom. This allowed me to cross off the lower left 3 and select the other 3 (and thus the 7 and 16)

#1 #21 #9#12 X3 #16#7 #18 X23 #14 X19 X5 #27#20 X32 #17 #25 #4 #11

Here the log of initial choices exists. #1 means it was chosen, (and if two exist, the other is ruled out), X3 means it was ruled out (and if two exist the other was chosen), and two parts written together were chosen together.

The path can't go through the 31 on the right as it would have to go through both 28 or the other 31 if it did.

X31 X2 #16 x13

Suppose the bottom 28 is in the path, and further suppose the top 30 is in the path. Then one can come up with these two paths.

`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 01# | 16X | 05#---20#---25#---09#---21# | 01X |`
`     +  |  +-----+  |  +-----+-----+-----+  |  +-----+`
`     | 18# | 10#---27# | 26x | 11x | 17#---12# | 32X |`
`     +  |  +  |  +-----+-----+-----+  |  +-----+-----+`
`     | 32#---11# | 15#---19# | 08#---06# | 27X | 20X |`
`     +-----+-----+  |  +  |  +  |  +-----+-----+-----+`
`     | 17x | 04\ | 13# | 24#---30# | 28x | 31X | 02X |`
`     +-----+-----+  |  +-----+-----+-----+-----+-----+`
`     | 25X | 10\ | 02#---26#---04----28# | 22x | 13x |`
`     +-----+-----+-----+-----+-----+  |  +-----+-----+`
`     | 05X | 14X | 30x | 08x | 15x | 31# | 19x | 06x |`
`     +-----+-----+-----+-----+-----+  |  +-----+-----+`
`     | 23X | 07X | 24x | 16#---29#---22# | 18X | 19X |`
`     +-----+-----+-----+  |  +-----+-----+-----+-----+`
`     | 03X | 12X | 09X | 03#---07#---14#---23#---33# |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`    +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 01# | 16X | 05#---20#---25#---09#---21# | 01X |`
`     +  |  +-----+  |  +-----+-----+-----+  |  +-----+`
`     | 18# | 10#---27# | 26x | 11x | 17#---12# | 32X |`
`     +  |  +  |  +-----+-----+-----+  |  +-----+-----+`
`     | 32#---11# | 15_ | 19_ | 08#---06# | 27X | 20X |`
`     +-----+-----+-----+-----+  |  +-----+-----+-----+`
`     | 17x | 04\ | 13#---24#---30# | 28x | 31X | 02X |`
`     +-----+-----+  |  +-----+-----+-----+-----+-----+`
`     | 25X | 10\ | 02#---26#---04----28#---22# | 13x |`
`     +-----+-----+-----+-----+-----+-----+  |  +-----+`
`     | 05X | 14X | 30x | 08x | 15#---31#---19# | 06x |`
`     +-----+-----+-----+-----+  |  +-----+-----+-----+`
`     | 23X | 07X | 24x | 16#---29# | 22_ | 18X | 19X |`
`     +-----+-----+-----+  |  +-----+-----+-----+-----+`
`     | 03X | 12X | 09X | 03#---07#---14#---23#---33# |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`

Suppose the lower 30 is included. Thus, the bottom 24 must be included, and thus the top 15 must also be included in this case. However then, the 13, 26, and 30 all need to connect with the 02, and problems also exist with the 29, 22, 7, and 14 on the bottom. So this case gives no possible solutions.

(no solution)

`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 01# | 16X | 05#---20#---25#---09#---21# | 01X |`
`     +  |  +-----+  |  +-----+-----+-----+  |  +-----+`
`     | 18# | 10#---27# | 26x | 11x | 17#---12# | 32X |`
`     +  |  +  |  +-----+-----+-----+  |  +-----+-----+`
`     | 32#---11# | 15#---19#---08#---06# | 27X | 20X |`
`     +-----+-----+  |  +-----+-----+-----+-----+-----+`
`     | 17x | 04x | 13# | 24x | 30x | 28x | 31X | 02X |`
`     +-----+-----+  |  +-----+-----+-----+-----+-----+`
`     | 25X | 10x | 02#---26#---04#---28# | 22x | 13x |`
`     +-----+-----+  |  +-----+-----+  |  +-----+-----+`
`     | 05X | 14X | 30# | 08x | 15x | 31# | 19x | 06x |`
`     +-----+-----+  |  +-----+-----+  |  +-----+-----+`
`     | 23X | 07X | 24#---16# | 29#---22# | 18X | 19X |`
`     +-----+-----+-----+  |  +  |  +  |  +-----+-----+`
`     | 03X | 12X | 09X | 03#---07#---14#---23#---33# |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`

Next consider if the top 28 is included. This means the 22 and 19 on the right side aren't in the path, and thus, the path must go through the other 19 and 22. Further, it must go through the bottom 15 from the 31. This prevents it from going through the top 15.

Supposing the bottom 10 is chosen, this causes the top 4, top 26, and top 30 to be chosen, thus preventing the path from being able to continue to the bottom of the grid.

(no solution)

`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 01# | 16X | 05#---20#---25#---09#---21# | 01X |`
`     +  |  +-----+  |  +-----+-----+-----+  |  +-----+`
`     | 18# | 10\ | 27#---26# | 11x | 17#---12# | 32X |`
`     +  |  +-----+-----+  |  +-----+  |  +-----+-----+`
`     | 32#---11# | 15x | 19# | 08_ | 06# | 27X | 20X |`
`     +-----+  |  +-----+-----+-----+  |  +-----+-----+`
`     | 17x | 04# | 13#---24# | 30#---28# | 31X | 02X | `
`     +-----+  |  +  |  +-----+-----+-----+-----+-----+`
`     | 25X | 10#---02# | 26x | 04x | 28x | 22x | 13x |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 05X | 14X | 30x | 08_ | 15#---31# | 19x | 06x |`
`     +-----+-----+-----+-----+-----+  |  +-----+-----+`
`     | 23X | 07X | 24x | 16# | 29# | 22# | 18X | 19X |`
`     +-----+-----+-----+  |  +-----+-----+-----+-----+`
`     | 03X | 12X | 09X | 03#---07# | 14#---23#---33# |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`

When the top 10 is chosen, the whole grid can be filled in, ending by choosing the bottom 15 and top 8. This results in another solution.

`     +-----+-----+-----+-----+-----+-----+-----+-----+`
`     | 01# | 16X | 05#---20#---25#---09#---21# | 01X |`
`     +  |  +-----+  |  +-----+-----+-----+  |  +-----+`
`     | 18# | 10#---27# | 26x | 11x | 17#---12# | 32X |`
`     +  |  +  |  +-----+-----+-----+  |  +-----+-----+`
`     | 32#---11# | 15x | 19#---08# | 06# | 27X | 20X |`
`     +-----+-----+-----+  |  +  |  +  |  +-----+-----+`
`     | 17x | 04x | 13#---24# | 30#---28# | 31X | 02X | `
`     +-----+-----+  |  +-----+-----+-----+-----+-----+`
`     | 25X | 10x | 02#---26#---04# | 28x | 22x | 13x |`
`     +-----+-----+-----+-----+  |  +-----+-----+-----+`
`     | 05X | 14X | 30x | 08x | 15#---31# | 19x | 06x |`
`     +-----+-----+-----+-----+-----+  |  +-----+-----+`
`     | 23X | 07X | 24x | 16#---29#---22# | 18X | 19X |`
`     +-----+-----+-----+  |  +-----+-----+-----+-----+`
`     | 03X | 12X | 09X | 03#---07# | 14#---23#---33# |`
`     +-----+-----+-----+-----+-----+-----+-----+-----+`

 Posted by Gamer on 2008-09-24 02:26:24

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