If $ a\equal{}2b\plus{}c$, $ b\equal{}2c\plus{}d$, $ 2c\equal{}d\plus{}a\minus{}1$, $ d\equal{}a\minus{}c$, what is $ b$?
1. Given the equations:
   \[
   a = 2b + c
   \]
   \[
   b = 2c + d
   \]
   \[
   2c = d + a - 1
   \]
   \[
   d = a - c
   \]

2. Substitute \( d = a - c \) into the second and third equations:
   \[
   b = 2c + (a - c) = c + a
   \]
   \[
   2c = (a - c) + a - 1 \implies 2c = 2a - c - 1 \implies 3c = 2a - 1
   \]

3. Now we have the simplified system:
   \[
   a = 2b + c
   \]
   \[
   b = c + a
   \]
   \[
   3c = 2a - 1
   \]

4. From \( b = c + a \), solve for \( c \):
   \[
   c = b - a
   \]

5. Substitute \( c = b - a \) into \( 3c = 2a - 1 \):
   \[
   3(b - a) = 2a - 1 \implies 3b - 3a = 2a - 1 \implies 3b = 5a - 1
   \]

6. Substitute \( c = b - a \) into \( a = 2b + c \):
   \[
   a = 2b + (b - a) \implies a = 2b + b - a \implies a = 3b - a \implies 2a = 3b \implies a = \frac{3b}{2}
   \]

7. Substitute \( a = \frac{3b}{2} \) into \( 3b = 5a - 1 \):
   \[
   3b = 5 \left( \frac{3b}{2} \right) - 1 \implies 3b = \frac{15b}{2} - 1 \implies 6b = 15b - 2 \implies 6b - 15b = -2 \implies -9b = -2 \implies b = \frac{2}{9}
   \]

The final answer is \( \boxed{b = \frac{2}{9}} \)