A car is travelling at a uniform speed. The driver sees a milestone showing a…

Concept: When a body moves at a uniform (constant) speed, it covers equal distances in equal time intervals. So if milestone readings are taken exactly one hour apart, the numeric gain from one reading to the next is constant, and that constant gain equals the speed itself.

1. Let the first milestone reading be a two-digit perfect square, written as 10a + b (a = tens digit, b = units digit). The two-digit perfect squares are 16, 25, 36, 49, 64, and 81.

2. After one hour, the reading becomes the digits reversed, 10b + a. Since the reading must increase (the car moves forward), this needs b > a, and the gain for that hour is (10b + a) − (10a + b) = 9(b − a).

3. Checking each two-digit perfect square with b > a: 16 reverses to 61 (gain 45), 25 reverses to 52 (gain 27), 36 reverses to 63 (gain 27), and 49 reverses to 94 (gain 45). (64 and 81 have b < a, so their reversal would be smaller — not possible for a forward-moving car, so they are ruled out.)

4. Apply the second-hour condition to all four surviving reversals by adding the same hourly gain again: 25 → 52 → 52 + 27 = 79, and 36 → 63 → 63 + 27 = 90 — both stay two-digit numbers, but the milestone after the second hour must be a three-digit number, so 25 and 36 are ruled out immediately. Only 16 → 61 → 61 + 45 = 106 and 49 → 94 → 94 + 45 = 139 cross into three digits, so these two remain the only candidates.

5. To choose between 106 and 139, apply the puzzle's standard reading of "the same two digits as part of a three-digit number": the original two digits reappear in the same order with a 0 inserted between them (digit-a, 0, digit-b) — for starting digits 1 and 6 that pattern is 106, and for starting digits 4 and 9 it would be 409. The actual second-hour reading for 16's sequence is 106, which matches the pattern exactly; the actual second-hour reading for 49's sequence is 139, which does not match 409 — so 49 is ruled out.

6. So the milestone sequence is 16 → 61 → 106, and the constant hourly gain — the car's speed — is 45 kmph.

Cross-check: 61 − 16 = 45 and 106 − 61 = 45 — both hourly gains match, and 16 = 4² is indeed a perfect square, confirming the uniform speed of 45 kmph.