The Mystery of the Missing Pages
Samantha was reading a book when she noticed that some pages were missing. She checked and found that the sum of the missing page numbers was 98.
If a sheet of paper in the book contains two pages (one on each side), how many sheets were missing?
Each sheet contains two consecutive page numbers, for example:
ReplyDeleteSheet 1: pages 1 and 2
Sheet 2: pages 3 and 4
Sheet 3: pages 5 and 6
…and so on.
Let’s suppose n sheets are missing. Then there are 2n page numbers missing.
We also know the sum of these missing page numbers is 98.
Let’s look at the sum of page numbers on each missing sheet:
Each sheet has two consecutive numbers: one odd, one even.
For example, Sheet 1: 1 + 2 = 3
Sheet 2: 3 + 4 = 7
Sheet 3: 5 + 6 = 11
…and so on.
So each sheet’s page sum increases by 4:
3, 7, 11, 15, …
That’s an arithmetic sequence:
Sum of an arithmetic sequence = (n/2) × (first term + last term)
We’re told the total sum is 98.
Try adding consecutive sheet sums until we reach 98:
3 (1st sheet)
3 + 7 = 10 (2 sheets)
10 + 11 = 21 (3 sheets)
21 + 15 = 36 (4 sheets)
36 + 19 = 55 (5 sheets)
55 + 23 = 78 (6 sheets)
78 + 27 = 105 (too much)
So try 6 sheets: 3 + 7 + 11 + 15 + 19 + 23 = 78 → not enough
Try 7 sheets: total is 105 → too much
Now check 98 – what group of 2-page numbers add to 98?
Try listing full sheet page pairs:
(1,2) → 3
(3,4) → 7
(5,6) → 11
(7,8) → 15
(9,10) → 19
(11,12) → 23
(13,14) → 27
(15,16) → 31
Total so far: 3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 = 136 (too high)
Now try a different approach:
Let’s try pairs that add up to 98:
Try grouping sheets and check sum:
Sheet 4: (7,8) → 15
Sheet 5: (9,10) → 19
Sheet 6: (11,12) → 23
Sheet 7: (13,14) → 27
Total = 15 + 19 + 23 + 27 = 84
Now add Sheet 8: (15,16) → 31 → total becomes 115 (too much)
Try removing sheet 4 (7+8=15), total becomes:
19 + 23 + 27 = 69
Try adding sheet 3 (5,6) → 11 → total: 80
Add sheet 9: (17,18) → 35 → 80 + 35 = 115 (too much)
Let's go back and solve algebraically.
Let the first page number on the first missing sheet be p.
Then the page numbers on the missing sheets are:
p, p+1
p+2, p+3
...
There are n sheets, so 2n pages.
So the page numbers are:
p, p+1, p+2, p+3, ..., p+2n–2, p+2n–1
These form an arithmetic sequence of 2n terms, starting from p.
Sum of arithmetic sequence:
Sum = (number of terms / 2) × (first + last)
= (2n / 2) × (p + (p + 2n – 1))
= n × (2p + 2n – 1) = 98
Now solve:
n × (2p + 2n – 1) = 98
Try small values of n:
Try n = 3:
3 × (2p + 6 – 1) = 98
3 × (2p + 5) = 98
2p + 5 = 98 ÷ 3 = not whole → skip
Try n = 4:
4 × (2p + 8 – 1) = 98
4 × (2p + 7) = 98
2p + 7 = 24.5 → not whole
Try n = 5:
5 × (2p + 10 – 1) = 98
5 × (2p + 9) = 98
2p + 9 = 19.6 → not whole
Try n = 6:
6 × (2p + 11) = 98
2p + 11 = 98 ÷ 6 = 16.333 → nope
Try n = 7:
7 × (2p + 13) = 98
2p + 13 = 14
2p = 1 → p = 0.5 → invalid
Try n = 2:
2 × (2p + 3) = 98
2p + 3 = 49
2p = 46 → p = 23
Page numbers: 23,24,25,26 → sum = 98 ✅
Final Answer:
2 sheets were missing.