The cryptic clues:
LIT - Burning books (ddef)
TRAIN - _T_ + IRAN*
ARK - AK + R (as in a real number)
The alphametic:
We know the dividend is TRAIN, but do not know yet which of ARK or LIT is the divisor. From the first subtraction the ones digit tells us that P is 0 and the hundreds digit tells us that T = O + 1, and that the tens digit must have needed to borrow. This tells us that 10+R = 2G, and thus R is even. Hence R is one of 2/4/6/8, and G is commensurately one of 6/7/8/9.
The last subtraction:
Here the tens digit, coupled with the fact that P=0, tells us that the ones digit must have needed a borrow, and that R = A+1. Now the ones digit tells us 10+N - L = O. Doesn't seem useful now, but I'm sure we'll need it. The alphametic thus far:
Chasing some equations to bust it open:
From the second subtraction, we know there must have been a borrow from the hundreds place, so we have G-1 - I = A. But we already have a relation between A and G: 10+R = 2G, and R = A+1, so we have 11+A = 2G. Thus we have G - 1 - I = 2G - 11, which is equivalent to I = 10 - G, hence I is one of 1/2/3/4. Listing cases we have the possible R/G/A/I values 2/6/1/4, 4/7/3/3, 6/8/5/2 and 8/9/7/1, and the duplicate 3 shows the second case cannot occur. But...there's another relationship from the second subtraction, in the ones digit, namely that either I - R = R, or 10 + I - R = R. This eliminates the last case, so we must have R/G/A/I as either 2/6/1/4 or 6/8/5/2.
So this forces A to be either 1 or 5. 5 seems unlikely, so let's show it's not. If A is 5, then R is 6, so the result of a last digit of the quotient times the divisor must be at least 5561. This forces the divisor to be greater than 617. But, the first and second digits of the quotient times the divisor are both less than 1000, and neither is 0. This cannot occur, as 617 times 2 is greater than 1000. Thus we must have R/G/A/I = 2/6/1/4. The alphametic thus far:
Finishing up:
We can now conclude that ARK must be the divisor, for if not one of the products in the division would be LIT, but this does not appear. We also know that T = O+1, forcing one of them to be even, from which we conclude that T/O is either 9/8 or 8/7. Moreover, the remaining digits K, L and N must all be odd. Looking at the last multiplication, we see that T times 12K is odd, as it ends in L, which forces T times K to be odd, which forces both T and K to be odd. Hence T is 9 and O is 8. The first multiplication quickly forces L to be 7, as 861 divided by 130 is greater than 6, and this forces K to be 3 and thus N to be 5. The finished alphametic:
The final answer:
is PARKING LOT, which ties into the title through the classic Joni Mitchell song Big Yellow Taxi, which includes the lyric "They paved paradise, put up a parking lot."
