Count to 100 Maze Printable - Teach Beside Me - Free Printable
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Step-by-step solution for: Count to 100 Maze Printable - Teach Beside Me
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Show Answer Key & Explanations
Step-by-step solution for: Count to 100 Maze Printable - Teach Beside Me
To solve this maze, we need to find a continuous path of numbers starting at 1 and ending at 100, moving only to adjacent circles (up, down, left, right — not diagonally), where each next number is exactly one more than the previous.
Let’s trace the path step by step from 1 to 100.
We start at the bottom-left corner:
→ The circle with 1 is at row 12, column 1 (if we count rows from top to bottom, columns left to right). But actually, looking at the grid layout:
The “START” arrow points to the bottom-left circle labeled 1.
The “FINISH” arrow points to the bottom-right circle labeled 100.
So we begin at 1 (bottom-left) and must reach 100 (bottom-right), following consecutive numbers.
Let’s map out the correct sequence by checking neighbors at each step.
I’ll go row by row from bottom up, since the start is at the bottom.
Row 12 (bottom row):
Positions (left to right):
1, 90, 1, 90, 87, 86, 93, 84, 95, 96, 99, 100 ← FINISH
But wait — there are two 1s in the bottom row? Let me reindex properly.
Actually, let’s label the grid as 12 rows × 12 columns.
Let me write the grid with row and column indices for clarity (row 1 = top, row 12 = bottom; col 1 = left, col 12 = right):
Row 1: 21, 21, 23, 24, 22, 27, 40, 30, 32, 32, 33, 34
Row 2: 22, 20, 22, 23, 28, 26, 28, 29, 30, 31, 35, 36
Row 3: 19, 21, 94, 24, 25, 27, 46, 40, 39, 36, 56, 57
Row 4: 18, 17, 16, 15, 48, 45, 44, 41, 38, 37, 55, 58
Row 5: 11, 32, 12, 14, 47, 46, 43, 42, 53, 54, 60, 59
Row 6: 61, 21, 15, 13, 12, 48, 49, 52, 63, 62, 61, 62
Row 7: 7, 1, 17, 14, 11, 12, 50, 51, 64, 73, 74, 75
Row 8: 82, 53, 42, 69, 8, 10, 66, 65, 72, 75, 76, 77
Row 9: 71, 4, 5, 6, 68, 9, 67, 70, 71, 80, 79, 78
Row 10: 3, 3, 4, 7, 7, 8, 68, 69, 82, 81, 82, 100
Row 11: 2, 4, 89, 88, 91, 92, 85, 94, 83, 95, 97, 98
Row 12: 1, 90, 1, 90, 87, 86, 93, 84, 95, 96, 99, 100
Wait — Row 12 has two 1s? That can’t be right for a unique path. Let me double-check the image description.
Actually, looking again: In Row 12 (bottom row), it's:
Col 1: 1 ← START
Col 2: 90
Col 3: 1 ← another 1?
Col 4: 90
...
Col 12: 100 ← FINISH
But if there are two 1s, which one is the real start? The START arrow points to the first 1 on the far left (col 1, row 12).
Also, note that in Row 7, Col 2: there’s a 1 too! So multiple 1s exist. We must follow the path that connects 1 → 2 → 3 → ... → 100 without skipping or repeating.
Let’s try tracing from the START (Row 12, Col 1 = 1):
From (12,1)=1, where can we go? Adjacent cells:
- Up: (11,1) = 2 → good! 1→2
- Right: (12,2) = 90 → too big, skip
- Down/Left: none (edge)
So: 1 → 2 (at 11,1)
Now at (11,1)=2
Adjacent:
- Up: (10,1)=3 → perfect! 2→3
- Right: (11,2)=4 → also possible, but let’s see what leads forward
- Down: (12,1)=1 ← already used
- Left: none
Try 2→3 (up to 10,1)
At (10,1)=3
Adjacent:
- Up: (9,1)=71 → too big
- Right: (10,2)=3 ← same number? Not increasing. Skip.
- Down: (11,1)=2 ← used
- Left: none
Wait — (10,2) is also 3. But we need 4 next.
From (10,1)=3, right is (10,2)=3 — not helpful.
What about from (11,1)=2, instead of going up to 3, go right to (11,2)=4? Then we’d have 1→2→4 — missing 3. Invalid.
Alternative: Maybe from (12,1)=1, we don’t go up to 2 immediately? But 2 is directly above — seems logical.
Wait — perhaps I made a mistake. Let’s look at Row 11, Col 1: it’s 2. Row 10, Col 1: it’s 3. Row 9, Col 1: 71 — too big.
But then from 3 at (10,1), how do we get to 4?
Look around (10,1)=3:
- Right: (10,2)=3 — no
- Up: (9,1)=71 — no
- Down: (11,1)=2 — used
- Left: none
Stuck? That suggests my initial move might be wrong.
Alternative idea: Maybe the path doesn’t go straight up from 1. What if from (12,1)=1, we go right to (12,2)=90? No, 90 is way too big.
Or maybe the other 1 in Row 12, Col 3 is part of the path? But START arrow points to Col 1.
Wait — let’s check Row 7, Col 2: it’s 1. And Row 8, Col 5: 8, etc. Perhaps the intended path starts at the 1 in Row 7, Col 2? But the START arrow clearly points to bottom-left 1.
Another approach: Let’s work backwards from 100.
FINISH is at (12,12)=100
Before 100 should be 99. Where is 99? At (12,11)=99 → yes! So 99→100
Before 99: 98. Where is 98? (11,12)=98 → so 98→99→100
Before 98: 97. (11,11)=97 → 97→98→99→100
Before 97: 96. (12,10)=96 → 96→97→98→99→100
Before 96: 95. (12,9)=95 OR (11,10)=95 → both exist. Let’s pick (12,9)=95 → then 95→96→...
Before 95: 94. (11,8)=94 OR (3,3)=94 — likely (11,8)=94 since closer.
(11,8)=94 → then 94→95→96→97→98→99→100
Before 94: 93. (12,7)=93 → 93→94→...
Before 93: 92. (11,6)=92 → 92→93→...
Before 92: 91. (11,5)=91 → 91→92→...
Before 91: 90. (12,4)=90 OR (12,2)=90 — say (12,4)=90 → 90→91→...
Before 90: 89. (11,3)=89 → 89→90→...
Before 89: 88. (11,4)=88 → 88→89→...
Before 88: 87. (12,5)=87 → 87→88→...
Before 87: 86. (12,6)=86 → 86→87→...
Before 86: 85. (11,7)=85 → 85→86→...
Before 85: 84. (12,8)=84 → 84→85→...
Before 84: 83. (11,9)=83 → 83→84→...
Before 83: 82. (10,9)=82 OR (8,1)=82 OR (10,11)=82 — let’s take (10,9)=82 → 82→83→...
Before 82: 81. (10,10)=81 → 81→82→...
Before 81: 80. (9,10)=80 → 80→81→...
Before 80: 79. (9,11)=79 → 79→80→...
Before 79: 78. (9,12)=78 → 78→79→...
Before 78: 77. (8,12)=77 → 77→78→...
Before 77: 76. (8,11)=76 → 76→77→...
Before 76: 75. (8,10)=75 OR (7,12)=75 — take (8,10)=75 → 75→76→...
Before 75: 74. (7,11)=74 → 74→75→...
Before 74: 73. (7,10)=73 → 73→74→...
Before 73: 72. (8,9)=72 → 72→73→...
Before 72: 71. (9,1)=71 OR (9,9)=71 — take (9,9)=71 → 71→72→...
Before 71: 70. (9,8)=70 → 70→71→...
Before 70: 69. (8,4)=69 OR (10,7)=69 — take (8,4)=69 → 69→70→...
Before 69: 68. (9,5)=68 OR (10,7)=68 — take (9,5)=68 → 68→69→...
Before 68: 67. (9,7)=67 → 67→68→...
Before 67: 66. (8,7)=66 → 66→67→...
Before 66: 65. (8,8)=65 → 65→66→...
Before 65: 64. (7,9)=64 → 64→65→...
Before 64: 63. (6,9)=63 → 63→64→...
Before 63: 62. (6,10)=62 OR (6,12)=62 — take (6,10)=62 → 62→63→...
Before 62: 61. (6,1)=61 OR (6,11)=61 — take (6,1)=61 → 61→62→...
Before 61: 60. (5,11)=60 → 60→61→...
Before 60: 59. (5,12)=59 → 59→60→...
Before 59: 58. (4,12)=58 → 58→59→...
Before 58: 57. (3,12)=57 → 57→58→...
Before 57: 56. (3,11)=56 → 56→57→...
Before 56: 55. (4,11)=55 → 55→56→...
Before 55: 54. (5,10)=54 → 54→55→...
Before 54: 53. (5,9)=53 → 53→54→...
Before 53: 52. (6,8)=52 → 52→53→...
Before 52: 51. (7,8)=51 → 51→52→...
Before 51: 50. (7,7)=50 → 50→51→...
Before 50: 49. (6,7)=49 → 49→50→...
Before 49: 48. (6,6)=48 OR (4,5)=48 — take (6,6)=48 → 48→49→...
Before 48: 47. (5,5)=47 → 47→48→...
Before 47: 46. (5,6)=46 OR (3,7)=46 — take (5,6)=46 → 46→47→...
Before 46: 45. (4,6)=45 → 45→46→...
Before 45: 44. (4,7)=44 → 44→45→...
Before 44: 43. (5,7)=43 → 43→44→...
Before 43: 42. (5,8)=42 OR (8,3)=42 — take (5,8)=42 → 42→43→...
Before 42: 41. (4,8)=41 → 41→42→...
Before 41: 40. (3,8)=40 OR (1,7)=40 — take (3,8)=40 → 40→41→...
Before 40: 39. (3,9)=39 → 39→40→...
Before 39: 38. (4,9)=38 → 38→39→...
Before 38: 37. (4,10)=37 → 37→38→...
Before 37: 36. (3,10)=36 OR (2,12)=36 — take (3,10)=36 → 36→37→...
Before 36: 35. (2,11)=35 → 35→36→...
Before 35: 34. (1,12)=34 → 34→35→...
Before 34: 33. (1,11)=33 → 33→34→...
Before 33: 32. (1,9)=32 OR (1,10)=32 — take (1,10)=32 → 32→33→...
Before 32: 31. (2,10)=31 → 31→32→...
Before 31: 30. (2,9)=30 OR (1,8)=30 — take (2,9)=30 → 30→31→...
Before 30: 29. (2,8)=29 → 29→30→...
Before 29: 28. (2,7)=28 OR (2,5)=28 — take (2,7)=28 → 28→29→...
Before 28: 27. (1,6)=27 OR (3,6)=27 — take (1,6)=27 → 27→28→...
Before 27: 26. (2,6)=26 → 26→27→...
Before 26: 25. (3,5)=25 → 25→26→...
Before 25: 24. (3,4)=24 OR (1,4)=24 — take (3,4)=24 → 24→25→...
Before 24: 23. (3,3)=94? Wait no — (3,3)=94, but we need 23. Look: (1,3)=23, (2,4)=23 — take (2,4)=23 → 23→24→...
Before 23: 22. (1,5)=22 OR (2,1)=22 OR (2,3)=22 — take (1,5)=22 → 22→23→...
Before 22: 21. (1,1)=21 OR (1,2)=21 OR (3,2)=21 — take (1,2)=21 → 21→22→...
Before 21: 20. (2,2)=20 → 20→21→...
Before 20: 19. (3,1)=19 → 19→20→...
Before 19: 18. (4,1)=18 → 18→19→...
Before 18: 17. (4,2)=17 → 17→18→...
Before 17: 16. (4,3)=16 → 16→17→...
Before 16: 15. (4,4)=15 OR (6,3)=15 — take (4,4)=15 → 15→16→...
Before 15: 14. (5,4)=14 OR (7,4)=14 — take (5,4)=14 → 14→15→...
Before 14: 13. (6,4)=13 → 13→14→...
Before 13: 12. (6,5)=12 OR (5,3)=12 OR (7,6)=12 — take (6,5)=12 → 12→13→...
Before 12: 11. (5,1)=11 OR (7,5)=11 — take (5,1)=11 → 11→12→...
Before 11: 10. (8,6)=10 → 10→11→...
Before 10: 9. (9,6)=9 → 9→10→...
Before 9: 8. (8,5)=8 OR (10,6)=8 — take (8,5)=8 → 8→9→...
Before 8: 7. (7,1)=7 OR (9,4)=7 OR (10,4)=7 OR (10,5)=7 — take (7,1)=7 → 7→8→...
Before 7: 6. (9,4)=6 → 6→7→...
Before 6: 5. (9,3)=5 → 5→6→...
Before 5: 4. (9,2)=4 OR (10,3)=4 OR (11,2)=4 — take (9,2)=4 → 4→5→...
Before 4: 3. (10,1)=3 OR (10,2)=3 — take (10,1)=3 → 3→4→...
Before 3: 2. (11,1)=2 → 2→3→...
Before 2: 1. Now, where is 1 before 2? We have (12,1)=1 and (7,2)=1 and (12,3)=1.
If we came from (11,1)=2, then previous should be 1 at (12,1) — which is the START!
Perfect! So the full path is:
Start at (12,1)=1
→ (11,1)=2
→ (10,1)=3
→ (9,2)=4
→ (9,3)=5
→ (9,4)=6
→ (7,1)=7 [wait, from (9,4)=6 to (7,1)=7? Not adjacent!]
Mistake here. From (9,4)=6, adjacent cells:
- Up: (8,4)=69 — too big
- Down: (10,4)=7 — yes! (10,4)=7 → so 6→7 via (10,4)
Then from (10,4)=7, to (8,5)=8? Not adjacent.
From (10,4)=7, adjacent:
- Up: (9,4)=6 — used
- Down: (11,4)=88 — too big
- Left: (10,3)=4 — used
- Right: (10,5)=7 — same number, not increasing
Stuck again.
Alternative: After (9,4)=6, go to (10,5)=7? (10,5)=7 — yes, and it's adjacent to (9,4)? Diagonally? No, only orthogonal.
(9,4) and (10,5) are diagonal — not allowed.
So from (9,4)=6, only orthogonal neighbors:
- (8,4)=69
- (10,4)=7 ← valid!
- (9,3)=5 — used
- (9,5)=68
So (10,4)=7 is the only choice.
Then from (10,4)=7, where to 8?
Neighbors of (10,4):
- (9,4)=6 — used
- (11,4)=88 — too big
- (10,3)=4 — used
- (10,5)=7 — same, not increasing
No 8 nearby. Problem.
Perhaps the path goes differently after 6.
Let’s backtrack.
After 5 at (9,3)=5, instead of going to (9,4)=6, go to (10,3)=4? But 4 is less than 5 — invalid.
Or from (9,3)=5, up to (8,3)=42 — too big.
Down to (10,3)=4 — decreasing.
Right to (9,4)=6 — only option.
So stuck at 7.
Unless... from (10,4)=7, we go to (8,5)=8? Not adjacent.
Wait — (8,5)=8 is at row 8, col 5. (10,4) is row 10, col 4. Not adjacent.
But (9,5)=68 — too big.
Another idea: Maybe the 7 at (7,1) is reached earlier.
Let’s try a different route from the beginning.
Start at (12,1)=1
→ (11,1)=2
→ (11,2)=4? Missing 3.
No.
From (12,1)=1, is there a 2 elsewhere? Only (11,1)=2.
Perhaps the path uses the 1 at (7,2).
Let me try starting from (7,2)=1.
From (7,2)=1, adjacent:
- Up: (6,2)=21 — too big
- Down: (8,2)=53 — too big
- Left: (7,1)=7 — too big
- Right: (7,3)=17 — too big
No 2 nearby. So not useful.
Back to bottom-left 1.
Another possibility: From (12,1)=1, go right to (12,2)=90? No.
Or perhaps the grid has a typo, but unlikely.
Let’s list all positions of key numbers.
Position of 1: (7,2), (12,1), (12,3)
Position of 2: (11,1)
Position of 3: (10,1), (10,2)
Position of 4: (9,2), (10,3), (11,2)
Position of 5: (9,3)
Position of 6: (9,4)
Position of 7: (7,1), (9,4) is 6, (10,4)=7, (10,5)=7, (9,4) is 6, etc.
From (9,4)=6, to (10,4)=7 — ok.
From (10,4)=7, to where is 8? (8,5)=8 — not adjacent.
But (10,6)=8 — yes! (10,6)=8, and (10,4) to (10,6) is not adjacent — skips (10,5).
(10,5)=7, so from (10,4)=7 to (10,5)=7 — same number, not allowed.
From (10,4)=7, no direct 8.
Unless we go from (9,4)=6 to (9,5)=68 — no.
Perhaps after 5 at (9,3)=5, we go to (10,3)=4 — but that's decreasing.
I think I found the error.
Let's look at Row 9: 71, 4, 5, 6, 68, 9, 67, 70, 71, 80, 79, 78
So (9,2)=4, (9,3)=5, (9,4)=6, (9,6)=9
From (9,4)=6, to (9,6)=9? Not consecutive.
But (9,5)=68 — too big.
From (9,4)=6, down to (10,4)=7 — ok.
Then from (10,4)=7, right to (10,5)=7 — same, not good.
Down to (11,4)=88 — no.
Up to (9,4)=6 — used.
Left to (10,3)=4 — used.
Dead end.
Unless the path goes from 6 to 7 via a different route.
What if from (9,4)=6, we go to (8,4)=69 — no.
Perhaps the 6 at (9,4) is not the only 6. Is there another 6? In Row 9, only one 6.
Another idea: Maybe after 5 at (9,3)=5, we go to (8,3)=42 — no.
Let's consider that from (10,1)=3, instead of going to (9,2)=4, go to (10,2)=3 — same number, not allowed.
Or from (11,1)=2, go to (11,2)=4 — missing 3.
So the only way to get 3 is from (10,1) or (10,2).
Assume we go 1->2->3 at (10,1)
Then from (10,1)=3, to (10,2)=3 — invalid.
To (9,1)=71 — invalid.
So perhaps the path is: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (10,2) ? But (10,2)=3, not 4.
(10,2)=3, (10,3)=4 — so from (10,1)=3 to (10,2)=3 — not good, but if we go to (10,3)=4 directly? Not adjacent.
Distance between (10,1) and (10,3) is 2 steps — not allowed.
Unless we go (10,1)=3 -> (11,2)=4? (11,2)=4, and (10,1) to (11,2) is diagonal — not allowed.
Orthogonal only.
So (10,1) to (11,1)=2 — used, or (10,2)=3 — same, or (9,1)=71 — no.
This suggests that the path may use the other 1.
Let me try starting from the 1 at (12,3).
From (12,3)=1, adjacent:
- Up: (11,3)=89 — too big
- Down: none
- Left: (12,2)=90 — too big
- Right: (12,4)=90 — too big
No 2 nearby. Useless.
So only viable start is (12,1)=1.
Perhaps the path goes: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (9,2) -> 5 (9,3) -> 6 (9,4) -> 7 (10,4) -> 8 (10,6) ? Not adjacent.
Wait, (10,6)=8, and (10,4)=7, with (10,5)=7 in between. If we could jump, but we can't.
Unless the path is: after 6 at (9,4), go to (8,5)=8? Not adjacent.
I recall that in some mazes, you can move to any adjacent cell, including those with the same number, but the instruction says "follow the numbers from 1 to 100", implying consecutive integers.
Perhaps there's a 7 at (8,5) and we reach it from elsewhere.
Let's list the path from 8 onwards to see if we can connect.
From earlier backward trace, we had a long chain from 8 to 100.
From 8 at (8,5)=8, to 9 at (9,6)=9 — are they adjacent? (8,5) and (9,6) are diagonal — not allowed.
(8,5)=8, adjacent cells:
- Up: (7,5)=11 — too big
- Down: (9,5)=68 — too big
- Left: (8,4)=69 — too big
- Right: (8,6)=10 — ah! (8,6)=10, but we need 9 first.
From 8, should go to 9.
Where is 9? (9,6)=9
Is (8,5) adjacent to (9,6)? No, diagonal.
(8,6)=10, so from 8 to 10 — missing 9.
Not good.
From (8,5)=8, to (7,5)=11 — no.
Perhaps 8 is at (10,6)=8.
From (10,6)=8, adjacent:
- Up: (9,6)=9 — yes! 8->9
- Down: (11,6)=92 — too big
- Left: (10,5)=7 — yes! 7->8
- Right: (10,7)=68 — too big
So if we have 7 at (10,5)=7, then to 8 at (10,6)=8, then to 9 at (9,6)=9.
Perfect!
So let's rebuild the path with this.
Start: (12,1)=1
-> (11,1)=2
-> (10,1)=3
-> ? to 4
From (10,1)=3, how to 4? (10,2)=3 — not good. (10,3)=4 — not adjacent.
But (11,2)=4 — and (10,1) to (11,2) is diagonal — not allowed.
Unless we go from (10,1)=3 to (10,2)=3 — same number, perhaps allowed? But the instruction is "follow the numbers from 1 to 100", which implies incrementing by 1 each step.
Typically in such mazes, you move to the next number, so same number is not progress.
Perhaps the path is: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (10,2) but (10,2)=3, not 4.
I think I found it.
Look at Row 10: 3, 3, 4, 7, 7, 8, 68, 69, 82, 81, 82, 100
So (10,1)=3, (10,2)=3, (10,3)=4, (10,4)=7, (10,5)=7, (10,6)=8
From (10,1)=3, if we go to (10,2)=3 — same, then to (10,3)=4 — but from (10,2) to (10,3) is adjacent, and 3 to 4 is good, but we had 3 at (10,1), then moved to (10,2)=3 (same), then to (10,3)=4.
Is moving to the same number allowed? The instruction doesn't prohibit it, but typically in number mazes, you move to the next number in sequence, so staying on the same number might not be considered "following" the sequence.
However, in this case, since there are duplicate numbers, perhaps it's allowed to traverse them as long as you eventually reach the next number.
But let's assume that we can move to adjacent cells even if the number is the same, as long as we are progressing towards 100.
So path:
(12,1)=1
-> (11,1)=2
-> (10,1)=3
-> (10,2)=3 (same number, but necessary to reach 4)
-> (10,3)=4
-> (9,2)=4? (9,2)=4, and (10,3) to (9,2) is diagonal — not allowed.
From (10,3)=4, adjacent:
- Up: (9,3)=5 — yes! 4->5
- Down: (11,3)=89 — too big
- Left: (10,2)=3 — used
- Right: (10,4)=7 — too big for now
So 4->5 at (9,3)=5
Then 5->6 at (9,4)=6
Then 6->7 at (10,4)=7 or (10,5)=7
Say (10,4)=7
Then 7->8 at (10,6)=8? Not adjacent; (10,4) to (10,6) has (10,5)=7 in between.
From (10,4)=7 to (10,5)=7 (same), then to (10,6)=8
So: 6->7 (10,4) ->7 (10,5) ->8 (10,6)
Then 8->9 at (9,6)=9 (up from (10,6))
Then 9->10 at (8,6)=10 (up from (9,6))
Then 10->11 at (7,5)=11? (8,6)=10, adjacent to (7,6)=50 — too big, (8,5)=8 — used, (8,7)=66 — too big, (9,6)=9 — used.
(7,5)=11, and (8,6) to (7,5) is diagonal — not allowed.
From (8,6)=10, to (7,6)=50 — no.
To (8,7)=66 — no.
Stuck.
From (9,6)=9, to (8,6)=10 — good.
From (8,6)=10, to (7,6)=50 — no.
To (8,5)=8 — used.
To (8,7)=66 — no.
To (9,6)=9 — used.
No 11 nearby.
But (7,5)=11, and (8,6) is not adjacent to (7,5).
Unless we go from (8,6)=10 to (7,6)=50 — no.
Perhaps 11 is at (5,1)=11 or (7,5)=11.
From (8,6)=10, no direct 11.
Let's list the path from 10 onwards from our backward trace.
Earlier, we had:
... -> 10 (8,6) -> 11 (7,5) ? Not adjacent.
In backward trace, we had 10 at (8,6), then 11 at (7,5) — but not adjacent.
Perhaps 11 is at (5,1), but far away.
Another position for 11: (7,5)=11, (5,1)=11
From (8,6)=10, to (7,6)=50 — no.
Perhaps the 10 is at (8,6), and 11 is at (7,5), but since not adjacent, maybe the path is different.
Let's accept that from (10,6)=8 -> (9,6)=9 -> (8,6)=10 -> then to (7,6)=50 — no.
I recall that in the grid, at (7,7)=50, (7,8)=51, etc.
Perhaps after 10, we go to 11 at (7,5), but how to get there from (8,6).
Unless we go from (8,6)=10 to (8,5)=8 — backtracking.
Not good.
Let's try a different strategy. Let's simulate the path from 1 to 20 manually with coordinates.
Define grid as G[row][col], row 1 to 12, col 1 to 12.
G[12][1] = 1 // start
G[11][1] = 2
G[10][1] = 3
G[10][2] = 3 // same, but let's use it to reach 4
G[10][3] = 4
G[9][3] = 5 // up from (10,3)
G[9][4] = 6 // right from (9,3)
G[10][4] = 7 // down from (9,4)
G[10][5] = 7 // right from (10,4) — same number
G[10][6] = 8 // right from (10,5)
G[9][6] = 9 // up from (10,6)
G[8][6] = 10 // up from (9,6)
G[8][5] = 8 // left from (8,6) — but 8 is less than 10, not good.
From G[8][6]=10, where is 11? G[7][5]=11, but not adjacent.
G[7][6]=50 — too big.
G[8][7]=66 — too big.
G[9][6]=9 — used.
No 11 adjacent to 10 at (8,6).
But G[7][5]=11, and G[8][6] is not adjacent to it.
However, G[7][5]=11 is adjacent to G[7][6]=50, G[6][5]=12, G[8][5]=8, G[7][4]=14.
From G[8][6]=10, if we could go to G[7][5]=11, but can't.
Perhaps the 10 is at a different location.
Another 10? Only at (8,6).
Let's look at G[7
Let’s trace the path step by step from 1 to 100.
We start at the bottom-left corner:
→ The circle with 1 is at row 12, column 1 (if we count rows from top to bottom, columns left to right). But actually, looking at the grid layout:
The “START” arrow points to the bottom-left circle labeled 1.
The “FINISH” arrow points to the bottom-right circle labeled 100.
So we begin at 1 (bottom-left) and must reach 100 (bottom-right), following consecutive numbers.
Let’s map out the correct sequence by checking neighbors at each step.
I’ll go row by row from bottom up, since the start is at the bottom.
Row 12 (bottom row):
Positions (left to right):
1, 90, 1, 90, 87, 86, 93, 84, 95, 96, 99, 100 ← FINISH
But wait — there are two 1s in the bottom row? Let me reindex properly.
Actually, let’s label the grid as 12 rows × 12 columns.
Let me write the grid with row and column indices for clarity (row 1 = top, row 12 = bottom; col 1 = left, col 12 = right):
Row 1: 21, 21, 23, 24, 22, 27, 40, 30, 32, 32, 33, 34
Row 2: 22, 20, 22, 23, 28, 26, 28, 29, 30, 31, 35, 36
Row 3: 19, 21, 94, 24, 25, 27, 46, 40, 39, 36, 56, 57
Row 4: 18, 17, 16, 15, 48, 45, 44, 41, 38, 37, 55, 58
Row 5: 11, 32, 12, 14, 47, 46, 43, 42, 53, 54, 60, 59
Row 6: 61, 21, 15, 13, 12, 48, 49, 52, 63, 62, 61, 62
Row 7: 7, 1, 17, 14, 11, 12, 50, 51, 64, 73, 74, 75
Row 8: 82, 53, 42, 69, 8, 10, 66, 65, 72, 75, 76, 77
Row 9: 71, 4, 5, 6, 68, 9, 67, 70, 71, 80, 79, 78
Row 10: 3, 3, 4, 7, 7, 8, 68, 69, 82, 81, 82, 100
Row 11: 2, 4, 89, 88, 91, 92, 85, 94, 83, 95, 97, 98
Row 12: 1, 90, 1, 90, 87, 86, 93, 84, 95, 96, 99, 100
Wait — Row 12 has two 1s? That can’t be right for a unique path. Let me double-check the image description.
Actually, looking again: In Row 12 (bottom row), it's:
Col 1: 1 ← START
Col 2: 90
Col 3: 1 ← another 1?
Col 4: 90
...
Col 12: 100 ← FINISH
But if there are two 1s, which one is the real start? The START arrow points to the first 1 on the far left (col 1, row 12).
Also, note that in Row 7, Col 2: there’s a 1 too! So multiple 1s exist. We must follow the path that connects 1 → 2 → 3 → ... → 100 without skipping or repeating.
Let’s try tracing from the START (Row 12, Col 1 = 1):
From (12,1)=1, where can we go? Adjacent cells:
- Up: (11,1) = 2 → good! 1→2
- Right: (12,2) = 90 → too big, skip
- Down/Left: none (edge)
So: 1 → 2 (at 11,1)
Now at (11,1)=2
Adjacent:
- Up: (10,1)=3 → perfect! 2→3
- Right: (11,2)=4 → also possible, but let’s see what leads forward
- Down: (12,1)=1 ← already used
- Left: none
Try 2→3 (up to 10,1)
At (10,1)=3
Adjacent:
- Up: (9,1)=71 → too big
- Right: (10,2)=3 ← same number? Not increasing. Skip.
- Down: (11,1)=2 ← used
- Left: none
Wait — (10,2) is also 3. But we need 4 next.
From (10,1)=3, right is (10,2)=3 — not helpful.
What about from (11,1)=2, instead of going up to 3, go right to (11,2)=4? Then we’d have 1→2→4 — missing 3. Invalid.
Alternative: Maybe from (12,1)=1, we don’t go up to 2 immediately? But 2 is directly above — seems logical.
Wait — perhaps I made a mistake. Let’s look at Row 11, Col 1: it’s 2. Row 10, Col 1: it’s 3. Row 9, Col 1: 71 — too big.
But then from 3 at (10,1), how do we get to 4?
Look around (10,1)=3:
- Right: (10,2)=3 — no
- Up: (9,1)=71 — no
- Down: (11,1)=2 — used
- Left: none
Stuck? That suggests my initial move might be wrong.
Alternative idea: Maybe the path doesn’t go straight up from 1. What if from (12,1)=1, we go right to (12,2)=90? No, 90 is way too big.
Or maybe the other 1 in Row 12, Col 3 is part of the path? But START arrow points to Col 1.
Wait — let’s check Row 7, Col 2: it’s 1. And Row 8, Col 5: 8, etc. Perhaps the intended path starts at the 1 in Row 7, Col 2? But the START arrow clearly points to bottom-left 1.
Another approach: Let’s work backwards from 100.
FINISH is at (12,12)=100
Before 100 should be 99. Where is 99? At (12,11)=99 → yes! So 99→100
Before 99: 98. Where is 98? (11,12)=98 → so 98→99→100
Before 98: 97. (11,11)=97 → 97→98→99→100
Before 97: 96. (12,10)=96 → 96→97→98→99→100
Before 96: 95. (12,9)=95 OR (11,10)=95 → both exist. Let’s pick (12,9)=95 → then 95→96→...
Before 95: 94. (11,8)=94 OR (3,3)=94 — likely (11,8)=94 since closer.
(11,8)=94 → then 94→95→96→97→98→99→100
Before 94: 93. (12,7)=93 → 93→94→...
Before 93: 92. (11,6)=92 → 92→93→...
Before 92: 91. (11,5)=91 → 91→92→...
Before 91: 90. (12,4)=90 OR (12,2)=90 — say (12,4)=90 → 90→91→...
Before 90: 89. (11,3)=89 → 89→90→...
Before 89: 88. (11,4)=88 → 88→89→...
Before 88: 87. (12,5)=87 → 87→88→...
Before 87: 86. (12,6)=86 → 86→87→...
Before 86: 85. (11,7)=85 → 85→86→...
Before 85: 84. (12,8)=84 → 84→85→...
Before 84: 83. (11,9)=83 → 83→84→...
Before 83: 82. (10,9)=82 OR (8,1)=82 OR (10,11)=82 — let’s take (10,9)=82 → 82→83→...
Before 82: 81. (10,10)=81 → 81→82→...
Before 81: 80. (9,10)=80 → 80→81→...
Before 80: 79. (9,11)=79 → 79→80→...
Before 79: 78. (9,12)=78 → 78→79→...
Before 78: 77. (8,12)=77 → 77→78→...
Before 77: 76. (8,11)=76 → 76→77→...
Before 76: 75. (8,10)=75 OR (7,12)=75 — take (8,10)=75 → 75→76→...
Before 75: 74. (7,11)=74 → 74→75→...
Before 74: 73. (7,10)=73 → 73→74→...
Before 73: 72. (8,9)=72 → 72→73→...
Before 72: 71. (9,1)=71 OR (9,9)=71 — take (9,9)=71 → 71→72→...
Before 71: 70. (9,8)=70 → 70→71→...
Before 70: 69. (8,4)=69 OR (10,7)=69 — take (8,4)=69 → 69→70→...
Before 69: 68. (9,5)=68 OR (10,7)=68 — take (9,5)=68 → 68→69→...
Before 68: 67. (9,7)=67 → 67→68→...
Before 67: 66. (8,7)=66 → 66→67→...
Before 66: 65. (8,8)=65 → 65→66→...
Before 65: 64. (7,9)=64 → 64→65→...
Before 64: 63. (6,9)=63 → 63→64→...
Before 63: 62. (6,10)=62 OR (6,12)=62 — take (6,10)=62 → 62→63→...
Before 62: 61. (6,1)=61 OR (6,11)=61 — take (6,1)=61 → 61→62→...
Before 61: 60. (5,11)=60 → 60→61→...
Before 60: 59. (5,12)=59 → 59→60→...
Before 59: 58. (4,12)=58 → 58→59→...
Before 58: 57. (3,12)=57 → 57→58→...
Before 57: 56. (3,11)=56 → 56→57→...
Before 56: 55. (4,11)=55 → 55→56→...
Before 55: 54. (5,10)=54 → 54→55→...
Before 54: 53. (5,9)=53 → 53→54→...
Before 53: 52. (6,8)=52 → 52→53→...
Before 52: 51. (7,8)=51 → 51→52→...
Before 51: 50. (7,7)=50 → 50→51→...
Before 50: 49. (6,7)=49 → 49→50→...
Before 49: 48. (6,6)=48 OR (4,5)=48 — take (6,6)=48 → 48→49→...
Before 48: 47. (5,5)=47 → 47→48→...
Before 47: 46. (5,6)=46 OR (3,7)=46 — take (5,6)=46 → 46→47→...
Before 46: 45. (4,6)=45 → 45→46→...
Before 45: 44. (4,7)=44 → 44→45→...
Before 44: 43. (5,7)=43 → 43→44→...
Before 43: 42. (5,8)=42 OR (8,3)=42 — take (5,8)=42 → 42→43→...
Before 42: 41. (4,8)=41 → 41→42→...
Before 41: 40. (3,8)=40 OR (1,7)=40 — take (3,8)=40 → 40→41→...
Before 40: 39. (3,9)=39 → 39→40→...
Before 39: 38. (4,9)=38 → 38→39→...
Before 38: 37. (4,10)=37 → 37→38→...
Before 37: 36. (3,10)=36 OR (2,12)=36 — take (3,10)=36 → 36→37→...
Before 36: 35. (2,11)=35 → 35→36→...
Before 35: 34. (1,12)=34 → 34→35→...
Before 34: 33. (1,11)=33 → 33→34→...
Before 33: 32. (1,9)=32 OR (1,10)=32 — take (1,10)=32 → 32→33→...
Before 32: 31. (2,10)=31 → 31→32→...
Before 31: 30. (2,9)=30 OR (1,8)=30 — take (2,9)=30 → 30→31→...
Before 30: 29. (2,8)=29 → 29→30→...
Before 29: 28. (2,7)=28 OR (2,5)=28 — take (2,7)=28 → 28→29→...
Before 28: 27. (1,6)=27 OR (3,6)=27 — take (1,6)=27 → 27→28→...
Before 27: 26. (2,6)=26 → 26→27→...
Before 26: 25. (3,5)=25 → 25→26→...
Before 25: 24. (3,4)=24 OR (1,4)=24 — take (3,4)=24 → 24→25→...
Before 24: 23. (3,3)=94? Wait no — (3,3)=94, but we need 23. Look: (1,3)=23, (2,4)=23 — take (2,4)=23 → 23→24→...
Before 23: 22. (1,5)=22 OR (2,1)=22 OR (2,3)=22 — take (1,5)=22 → 22→23→...
Before 22: 21. (1,1)=21 OR (1,2)=21 OR (3,2)=21 — take (1,2)=21 → 21→22→...
Before 21: 20. (2,2)=20 → 20→21→...
Before 20: 19. (3,1)=19 → 19→20→...
Before 19: 18. (4,1)=18 → 18→19→...
Before 18: 17. (4,2)=17 → 17→18→...
Before 17: 16. (4,3)=16 → 16→17→...
Before 16: 15. (4,4)=15 OR (6,3)=15 — take (4,4)=15 → 15→16→...
Before 15: 14. (5,4)=14 OR (7,4)=14 — take (5,4)=14 → 14→15→...
Before 14: 13. (6,4)=13 → 13→14→...
Before 13: 12. (6,5)=12 OR (5,3)=12 OR (7,6)=12 — take (6,5)=12 → 12→13→...
Before 12: 11. (5,1)=11 OR (7,5)=11 — take (5,1)=11 → 11→12→...
Before 11: 10. (8,6)=10 → 10→11→...
Before 10: 9. (9,6)=9 → 9→10→...
Before 9: 8. (8,5)=8 OR (10,6)=8 — take (8,5)=8 → 8→9→...
Before 8: 7. (7,1)=7 OR (9,4)=7 OR (10,4)=7 OR (10,5)=7 — take (7,1)=7 → 7→8→...
Before 7: 6. (9,4)=6 → 6→7→...
Before 6: 5. (9,3)=5 → 5→6→...
Before 5: 4. (9,2)=4 OR (10,3)=4 OR (11,2)=4 — take (9,2)=4 → 4→5→...
Before 4: 3. (10,1)=3 OR (10,2)=3 — take (10,1)=3 → 3→4→...
Before 3: 2. (11,1)=2 → 2→3→...
Before 2: 1. Now, where is 1 before 2? We have (12,1)=1 and (7,2)=1 and (12,3)=1.
If we came from (11,1)=2, then previous should be 1 at (12,1) — which is the START!
Perfect! So the full path is:
Start at (12,1)=1
→ (11,1)=2
→ (10,1)=3
→ (9,2)=4
→ (9,3)=5
→ (9,4)=6
→ (7,1)=7 [wait, from (9,4)=6 to (7,1)=7? Not adjacent!]
Mistake here. From (9,4)=6, adjacent cells:
- Up: (8,4)=69 — too big
- Down: (10,4)=7 — yes! (10,4)=7 → so 6→7 via (10,4)
Then from (10,4)=7, to (8,5)=8? Not adjacent.
From (10,4)=7, adjacent:
- Up: (9,4)=6 — used
- Down: (11,4)=88 — too big
- Left: (10,3)=4 — used
- Right: (10,5)=7 — same number, not increasing
Stuck again.
Alternative: After (9,4)=6, go to (10,5)=7? (10,5)=7 — yes, and it's adjacent to (9,4)? Diagonally? No, only orthogonal.
(9,4) and (10,5) are diagonal — not allowed.
So from (9,4)=6, only orthogonal neighbors:
- (8,4)=69
- (10,4)=7 ← valid!
- (9,3)=5 — used
- (9,5)=68
So (10,4)=7 is the only choice.
Then from (10,4)=7, where to 8?
Neighbors of (10,4):
- (9,4)=6 — used
- (11,4)=88 — too big
- (10,3)=4 — used
- (10,5)=7 — same, not increasing
No 8 nearby. Problem.
Perhaps the path goes differently after 6.
Let’s backtrack.
After 5 at (9,3)=5, instead of going to (9,4)=6, go to (10,3)=4? But 4 is less than 5 — invalid.
Or from (9,3)=5, up to (8,3)=42 — too big.
Down to (10,3)=4 — decreasing.
Right to (9,4)=6 — only option.
So stuck at 7.
Unless... from (10,4)=7, we go to (8,5)=8? Not adjacent.
Wait — (8,5)=8 is at row 8, col 5. (10,4) is row 10, col 4. Not adjacent.
But (9,5)=68 — too big.
Another idea: Maybe the 7 at (7,1) is reached earlier.
Let’s try a different route from the beginning.
Start at (12,1)=1
→ (11,1)=2
→ (11,2)=4? Missing 3.
No.
From (12,1)=1, is there a 2 elsewhere? Only (11,1)=2.
Perhaps the path uses the 1 at (7,2).
Let me try starting from (7,2)=1.
From (7,2)=1, adjacent:
- Up: (6,2)=21 — too big
- Down: (8,2)=53 — too big
- Left: (7,1)=7 — too big
- Right: (7,3)=17 — too big
No 2 nearby. So not useful.
Back to bottom-left 1.
Another possibility: From (12,1)=1, go right to (12,2)=90? No.
Or perhaps the grid has a typo, but unlikely.
Let’s list all positions of key numbers.
Position of 1: (7,2), (12,1), (12,3)
Position of 2: (11,1)
Position of 3: (10,1), (10,2)
Position of 4: (9,2), (10,3), (11,2)
Position of 5: (9,3)
Position of 6: (9,4)
Position of 7: (7,1), (9,4) is 6, (10,4)=7, (10,5)=7, (9,4) is 6, etc.
From (9,4)=6, to (10,4)=7 — ok.
From (10,4)=7, to where is 8? (8,5)=8 — not adjacent.
But (10,6)=8 — yes! (10,6)=8, and (10,4) to (10,6) is not adjacent — skips (10,5).
(10,5)=7, so from (10,4)=7 to (10,5)=7 — same number, not allowed.
From (10,4)=7, no direct 8.
Unless we go from (9,4)=6 to (9,5)=68 — no.
Perhaps after 5 at (9,3)=5, we go to (10,3)=4 — but that's decreasing.
I think I found the error.
Let's look at Row 9: 71, 4, 5, 6, 68, 9, 67, 70, 71, 80, 79, 78
So (9,2)=4, (9,3)=5, (9,4)=6, (9,6)=9
From (9,4)=6, to (9,6)=9? Not consecutive.
But (9,5)=68 — too big.
From (9,4)=6, down to (10,4)=7 — ok.
Then from (10,4)=7, right to (10,5)=7 — same, not good.
Down to (11,4)=88 — no.
Up to (9,4)=6 — used.
Left to (10,3)=4 — used.
Dead end.
Unless the path goes from 6 to 7 via a different route.
What if from (9,4)=6, we go to (8,4)=69 — no.
Perhaps the 6 at (9,4) is not the only 6. Is there another 6? In Row 9, only one 6.
Another idea: Maybe after 5 at (9,3)=5, we go to (8,3)=42 — no.
Let's consider that from (10,1)=3, instead of going to (9,2)=4, go to (10,2)=3 — same number, not allowed.
Or from (11,1)=2, go to (11,2)=4 — missing 3.
So the only way to get 3 is from (10,1) or (10,2).
Assume we go 1->2->3 at (10,1)
Then from (10,1)=3, to (10,2)=3 — invalid.
To (9,1)=71 — invalid.
So perhaps the path is: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (10,2) ? But (10,2)=3, not 4.
(10,2)=3, (10,3)=4 — so from (10,1)=3 to (10,2)=3 — not good, but if we go to (10,3)=4 directly? Not adjacent.
Distance between (10,1) and (10,3) is 2 steps — not allowed.
Unless we go (10,1)=3 -> (11,2)=4? (11,2)=4, and (10,1) to (11,2) is diagonal — not allowed.
Orthogonal only.
So (10,1) to (11,1)=2 — used, or (10,2)=3 — same, or (9,1)=71 — no.
This suggests that the path may use the other 1.
Let me try starting from the 1 at (12,3).
From (12,3)=1, adjacent:
- Up: (11,3)=89 — too big
- Down: none
- Left: (12,2)=90 — too big
- Right: (12,4)=90 — too big
No 2 nearby. Useless.
So only viable start is (12,1)=1.
Perhaps the path goes: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (9,2) -> 5 (9,3) -> 6 (9,4) -> 7 (10,4) -> 8 (10,6) ? Not adjacent.
Wait, (10,6)=8, and (10,4)=7, with (10,5)=7 in between. If we could jump, but we can't.
Unless the path is: after 6 at (9,4), go to (8,5)=8? Not adjacent.
I recall that in some mazes, you can move to any adjacent cell, including those with the same number, but the instruction says "follow the numbers from 1 to 100", implying consecutive integers.
Perhaps there's a 7 at (8,5) and we reach it from elsewhere.
Let's list the path from 8 onwards to see if we can connect.
From earlier backward trace, we had a long chain from 8 to 100.
From 8 at (8,5)=8, to 9 at (9,6)=9 — are they adjacent? (8,5) and (9,6) are diagonal — not allowed.
(8,5)=8, adjacent cells:
- Up: (7,5)=11 — too big
- Down: (9,5)=68 — too big
- Left: (8,4)=69 — too big
- Right: (8,6)=10 — ah! (8,6)=10, but we need 9 first.
From 8, should go to 9.
Where is 9? (9,6)=9
Is (8,5) adjacent to (9,6)? No, diagonal.
(8,6)=10, so from 8 to 10 — missing 9.
Not good.
From (8,5)=8, to (7,5)=11 — no.
Perhaps 8 is at (10,6)=8.
From (10,6)=8, adjacent:
- Up: (9,6)=9 — yes! 8->9
- Down: (11,6)=92 — too big
- Left: (10,5)=7 — yes! 7->8
- Right: (10,7)=68 — too big
So if we have 7 at (10,5)=7, then to 8 at (10,6)=8, then to 9 at (9,6)=9.
Perfect!
So let's rebuild the path with this.
Start: (12,1)=1
-> (11,1)=2
-> (10,1)=3
-> ? to 4
From (10,1)=3, how to 4? (10,2)=3 — not good. (10,3)=4 — not adjacent.
But (11,2)=4 — and (10,1) to (11,2) is diagonal — not allowed.
Unless we go from (10,1)=3 to (10,2)=3 — same number, perhaps allowed? But the instruction is "follow the numbers from 1 to 100", which implies incrementing by 1 each step.
Typically in such mazes, you move to the next number, so same number is not progress.
Perhaps the path is: 1 (12,1) -> 2 (11,1) -> 3 (10,1) -> 4 (10,2) but (10,2)=3, not 4.
I think I found it.
Look at Row 10: 3, 3, 4, 7, 7, 8, 68, 69, 82, 81, 82, 100
So (10,1)=3, (10,2)=3, (10,3)=4, (10,4)=7, (10,5)=7, (10,6)=8
From (10,1)=3, if we go to (10,2)=3 — same, then to (10,3)=4 — but from (10,2) to (10,3) is adjacent, and 3 to 4 is good, but we had 3 at (10,1), then moved to (10,2)=3 (same), then to (10,3)=4.
Is moving to the same number allowed? The instruction doesn't prohibit it, but typically in number mazes, you move to the next number in sequence, so staying on the same number might not be considered "following" the sequence.
However, in this case, since there are duplicate numbers, perhaps it's allowed to traverse them as long as you eventually reach the next number.
But let's assume that we can move to adjacent cells even if the number is the same, as long as we are progressing towards 100.
So path:
(12,1)=1
-> (11,1)=2
-> (10,1)=3
-> (10,2)=3 (same number, but necessary to reach 4)
-> (10,3)=4
-> (9,2)=4? (9,2)=4, and (10,3) to (9,2) is diagonal — not allowed.
From (10,3)=4, adjacent:
- Up: (9,3)=5 — yes! 4->5
- Down: (11,3)=89 — too big
- Left: (10,2)=3 — used
- Right: (10,4)=7 — too big for now
So 4->5 at (9,3)=5
Then 5->6 at (9,4)=6
Then 6->7 at (10,4)=7 or (10,5)=7
Say (10,4)=7
Then 7->8 at (10,6)=8? Not adjacent; (10,4) to (10,6) has (10,5)=7 in between.
From (10,4)=7 to (10,5)=7 (same), then to (10,6)=8
So: 6->7 (10,4) ->7 (10,5) ->8 (10,6)
Then 8->9 at (9,6)=9 (up from (10,6))
Then 9->10 at (8,6)=10 (up from (9,6))
Then 10->11 at (7,5)=11? (8,6)=10, adjacent to (7,6)=50 — too big, (8,5)=8 — used, (8,7)=66 — too big, (9,6)=9 — used.
(7,5)=11, and (8,6) to (7,5) is diagonal — not allowed.
From (8,6)=10, to (7,6)=50 — no.
To (8,7)=66 — no.
Stuck.
From (9,6)=9, to (8,6)=10 — good.
From (8,6)=10, to (7,6)=50 — no.
To (8,5)=8 — used.
To (8,7)=66 — no.
To (9,6)=9 — used.
No 11 nearby.
But (7,5)=11, and (8,6) is not adjacent to (7,5).
Unless we go from (8,6)=10 to (7,6)=50 — no.
Perhaps 11 is at (5,1)=11 or (7,5)=11.
From (8,6)=10, no direct 11.
Let's list the path from 10 onwards from our backward trace.
Earlier, we had:
... -> 10 (8,6) -> 11 (7,5) ? Not adjacent.
In backward trace, we had 10 at (8,6), then 11 at (7,5) — but not adjacent.
Perhaps 11 is at (5,1), but far away.
Another position for 11: (7,5)=11, (5,1)=11
From (8,6)=10, to (7,6)=50 — no.
Perhaps the 10 is at (8,6), and 11 is at (7,5), but since not adjacent, maybe the path is different.
Let's accept that from (10,6)=8 -> (9,6)=9 -> (8,6)=10 -> then to (7,6)=50 — no.
I recall that in the grid, at (7,7)=50, (7,8)=51, etc.
Perhaps after 10, we go to 11 at (7,5), but how to get there from (8,6).
Unless we go from (8,6)=10 to (8,5)=8 — backtracking.
Not good.
Let's try a different strategy. Let's simulate the path from 1 to 20 manually with coordinates.
Define grid as G[row][col], row 1 to 12, col 1 to 12.
G[12][1] = 1 // start
G[11][1] = 2
G[10][1] = 3
G[10][2] = 3 // same, but let's use it to reach 4
G[10][3] = 4
G[9][3] = 5 // up from (10,3)
G[9][4] = 6 // right from (9,3)
G[10][4] = 7 // down from (9,4)
G[10][5] = 7 // right from (10,4) — same number
G[10][6] = 8 // right from (10,5)
G[9][6] = 9 // up from (10,6)
G[8][6] = 10 // up from (9,6)
G[8][5] = 8 // left from (8,6) — but 8 is less than 10, not good.
From G[8][6]=10, where is 11? G[7][5]=11, but not adjacent.
G[7][6]=50 — too big.
G[8][7]=66 — too big.
G[9][6]=9 — used.
No 11 adjacent to 10 at (8,6).
But G[7][5]=11, and G[8][6] is not adjacent to it.
However, G[7][5]=11 is adjacent to G[7][6]=50, G[6][5]=12, G[8][5]=8, G[7][4]=14.
From G[8][6]=10, if we could go to G[7][5]=11, but can't.
Perhaps the 10 is at a different location.
Another 10? Only at (8,6).
Let's look at G[7
Parent Tip: Review the logic above to help your child master the concept of preschool number 1 maze.