解:解方程$\frac{1 - 2x}{6} + \frac{x + 1}{3} = 1 - \frac{2x + 1}{4},$
去分母,得$2(1 - 2x)+4(x + 1)=12 - 3(2x + 1),$
去括号,得$2 - 4x + 4x + 4 = 12 - 6x - 3,$
移项,得$-4x + 4x + 6x = 12 - 3 - 2 - 4,$
合并同类项,得$6x = 3,$
系数化为$1,$得$x = \frac{1}{2}。$
解方程$x+\frac{6x - a}{3}=\frac{a}{6}-3x,$
去分母,得$6x + 2(6x - a)=a - 18x,$
去括号,得$6x + 12x - 2a = a - 18x,$
移项,得$6x + 12x + 18x = a + 2a,$
合并同类项,得$36x = 3a,$
系数化为$1,$得$x = \frac{1}{12}a。$
根据题意,得$\frac{1}{2}×\frac{1}{12}a = 1,$
$\frac{1}{24}a = 1,$
解得$a = 24。$
所以$(23 - a)^{203}=(23 - 24)^{203}=(-1)^{203}=-1$