Note that a temperature T converts back to T if and only if T + 9 converts
back to T + 9. Thus it is only necessary to examine nine consecutive
temperatures. It is easy to show that 32 converts back to 32, 33 and 34
both convert back to 34, 35 and 36 both convert back to 36, 37 and 38
both convert back to 37, and 39 and 40 both convert back to 39. Hence out
of every nine consecutive temperatures starting with 32, five are converted
correctly and four are not. For 32 ≤ T < 32 + 9 · 107 = 995. There are
107 ·5 = 535 temperatures that are converted correctly. The remaining six
temperatures 995, 996, . . . , 1000 behave like 32, 33, . . . , 37, so four of the
remaining six temperatures are converted correctly. Thus there is a total
of 535 + 4 = 539 temperatures.