Question

Driving along, terry notices that the last four digits on his car's odometer are palindromic. a mile later, the last five digits are palindromic. after driving another mile, the middle four digits are palindromic. one more mile, and all six are palindromic. what was the odometer reading when terry first looked at it? form a linear system of equations that expresses the requirements of this puzzle.

Answer 1

Designate the initial digits, left to right, as {a, b, c, d, d, c}.

After adding one mile, the digits are {a, b, c, d, c, b}, so the relevant equation is
10c +b = 10d +c +1

After adding another mile, the digits are {a, b, c, c, b, e}, so the relevant equation is
100c +10b +e = 100d +10c +b +1

After another mile, the digits are {a, b, c, c, b, a}, so the relevant equation is
a = e +1

In summary, we have 3 equations in 5 unknowns.
b + 9c -10d = 1
9b +90c -100d +e = 1
a - e = 1
along with the constraints {a, b, c, d, e} ∈ {0, ..., 9}


_____
These have the solution {a, b, c, d, e} = {1, 9, 8, 8, 0}, so the odometer readings were
198888
198889
198890
198891