Pretend there is a page 0 in the book. Since 0 is not 1, it won't
change the answer.
Let's look only at the 1000 page numbers from 0 to 999
Think of them all as if they were all three-digit numbers,
000-999
So if the pages were numbered 000 through 999, there would be
3000 digits (1000 hundreds digits, 1000 tens digits and 1000 ones
digits)and one tenth of them would be 0's, one tenths of
them 1's, one-tenth of them 2's,... and one tenth of them 9's.
So there are one-tenth of 3000 1's which amounts to 300 1's
on the pages from 000 through 999.
Let's determine how many 1's there are from pages 1000 through 1099
There are as many pages from 1000 through 1099 as there are from
00 through 99.
The number of digits from 00 through 99 is 200, 1/10 of them are
0's, 1/10 of them are 1's, 1/10 of them are 2's,..., 1/10 of them
are 9's. Since 1/10 of 200 is 20, there are 20 1's among the pages
00 through 99. And since there are 100 pages from 1000 thru 1099,
and each one has a 1 for its thousands digit, there are 20+100 or 120
1's from pages 1000 through 1099.
So far we have accounted for 120 1's from page 1000 through page 1099.
That's only 300+120 or 420 1's. We've still got 689-420=269 more 1's
to go. So we've ruled out answer a) 1024.
--
Now let's determine how many 1's there are from pages 1100 through 1199
There are as many pages from 1100 through 1199 as there are from
pages 00 through 99.
We have already determined above that there are 20 1's among the pages
00 through 99. And since there are 100 pages from 1100 thru 1199,
and each one has 1's for both its thousands digit and it's hundreds
digit, there are 20+100+100 or 220 1's from pages 1100 through 1199.
So far we have accounted for 220 1's from page 1100 through page 1199.
That's only 420+220 or 640 1's. We've still got 689-640=49 more 1's
to go with pages 1200 on up. So we've ruled out answer a) 1024, b) 1124,
and c) 1134, since they are less than 1199.
So the answer is either 1224 or 1234.
From page 1200 thru 1224, there are 25 1's for thousands digits,
the pages 1210 through 1224 have 10 1's for tens digits, and pages 1201,
1211, and 1221 each have an extra 1 for the ones digit, making 3 more
1's. So that's 25+10+3 or 38 1's from 1200 through 1224.
The total of 1's from page 0 through page 1224 is then 640+38 = 678.
We still have 689-678=11 more 1's to go. So we have ruled out answer
c) and so we now know the answer can only be d) 1234.
We could stop there, but let's see if it does actually work out to
be 1234.
1225,1226,1227,1228,1229,1230,1231,1232,1233,1234.
Count them and you'll see there are 11 more 1's from pages
1225 through 1234.
Answer: c) 1234.
Edwin
