256 lines
7.8 KiB
Plaintext
256 lines
7.8 KiB
Plaintext
I thought about a way to drastically reduce the number of necessary trees. Does it make sense ?
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For a 5 leaves (numbers) and 4 operators, I define following tree types:
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;; Type A
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o
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/ \
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o
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/ \
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o
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/ \
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o
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/ \
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;; type B
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o
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/ \
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o o
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/ \ / \
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o
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/ \
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;; Type C
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o
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/ \
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o
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/ \
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o o
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/ \ / \
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Normally, we have 14 trees for 5 numbers. But I categorize them within the 3 types above.
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What I think is: 2 trees of same type will find the same solutions, given we permute all operators and numbers :
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- It looks trivial for operators '+' and '-'.
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- For '/' et '-', if we swap subtrees numbers to always have maximum on the left, we are left with always one operation (if possible), so the order does not matter.
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My generated trees generated (in tree.c) for 5 numbers are :
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1: (Op x (Op x (Op x (Op x x))))
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2: (Op x (Op x (Op (Op x x) x)))
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3: (Op x (Op (Op x x) (Op x x)))
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4: (Op x (Op (Op x (Op x x)) x))
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5: (Op x (Op (Op (Op x x) x) x))
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6: (Op (Op x x) (Op x (Op x x)))
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7: (Op (Op x x) (Op (Op x x) x))
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8: (Op (Op x (Op x x)) (Op x x))
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9: (Op (Op x (Op x (Op x x))) x)
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10: (Op (Op x (Op (Op x x) x)) x)
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11: (Op (Op (Op x x) x) (Op x x))
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12: (Op (Op (Op x x) (Op x x)) x)
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13: (Op (Op (Op x (Op x x)) x) x)
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14: (Op (Op (Op (Op x x) x) x) x)
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Categories:
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;; 1 : type A
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(Op x o
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(Op x / \
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(Op x o
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(Op x / \
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x)))) o
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/ \
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o
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/ \
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;; 2 : Type A
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(Op x o
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(Op x / \
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(Op o
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(Op x / \
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x) o
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x))) / \
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o
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/ \
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;; 3 : Type C
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(Op x o
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(Op / \
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(Op x o
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x) / \
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(Op x o o
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x))) / \ / \
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;; 4 : Type A
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o
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(Op x / \
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(Op o
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(Op x / \
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(Op x o
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x)) / \
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x)) o
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/ \
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;; 5 : Type A
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(Op x o
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(Op / \
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(Op o
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(Op x / \
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x) o
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x) / \
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x)) o
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/ \
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;; 6 : Type B
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(Op o
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(Op x / \
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x) o o
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(Op x / \ / \
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(Op x o
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x))) / \
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;; 7 : Type B
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(Op o
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(Op x / \
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x) o o
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(Op / \ / \
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(Op x o
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x) / \
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x))
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;; 8 : Type B
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(Op o
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(Op x / \
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(Op x o o
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x)) / \ / \
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(Op x o
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x)) / \
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;; 9 : Type A
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(Op o
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(Op x / \
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(Op x o
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(Op x / \
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x))) o
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x) / \
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o
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/ \
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;; 10 : Type A
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(Op o
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(Op x / \
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(Op o
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(Op x / \
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x) o
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x)) / \
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x) o
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/ \
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;; 11 : Type B
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(Op o
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(Op / \
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(Op x o o
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x) / / \
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x) o
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(Op x / \
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x))
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;; 12 : Type C
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(Op o
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(Op / \
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(Op x o
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x) / \
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(Op x o o
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x)) / \ / \
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x)
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;; 13 : Type A
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(Op o
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(Op / \
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(Op x o
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(Op x / \
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x)) o
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x) / \
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x) o
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/ \
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;; 14 : Type A
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(Op o
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(Op / \
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(Op o
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(Op x / \
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x) o
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x) / \
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x) o
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x) / \
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Below are all categories by input of N numbers, for N={2,3,4,5,6} :
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w is max leaves on same level, and h is height (distance from root to lowest leaf).
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2 leaves (1, Catalan=1):
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o
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/ \
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width=2, height=1
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serialization: 100
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3 leaves (1, Catalan=2):
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o
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/ \
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o
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/ \
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w=2, h=2
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serial: 10100
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4 leaves (2, Catalan=5):
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o o
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/ \ / \
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o o o
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/ \ / \ / \
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o
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/ \
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w=2, h=3 w=4, h=2
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serial: 1010100 serial: 1100100
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5 leaves (3, Catalan=14):
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o o o
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/ \ / \ / \
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o o o o
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/ \ / \ / \ / \
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o o o o
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/ \ / \ / \ / \
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o
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/ \
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w=2, h=4 w=3, h=3 w=4, h=3
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serial: 101010100 serial: 110010100 serial: 101100100
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6 leaves (6, Catalan=42):
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o o o o
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/ \ / \ / \ / \
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o o o o o
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/ \ / \ / \ / \ / \
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o o o o o
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/ \ / \ / \ / \ / \
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o o o o o
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/ \ / \ / \ / \ / \
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o
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/ \
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w=2, h=5 w=3, h=4 w=3, h=4 w=4, h=4
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serial: 10101010100 serial: 11001010100 serial: 10110010100 serial: 10101100100
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o o
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/ \ / \
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o o o o
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/ \ / \ / \ / \
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o o o o
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/ \ / \ / \ / \
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w=4, h=3 w=4, h=3
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serial: 11001100100 serial: 11010010100
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