Missing pages
In a book, few consecutive pages are missing. The sum of page numbers of missing pages is 1072. How many sheets of paper are missing?
The answer is 16.
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2 solutions
I came up with the same answer using a simple Excel formula. I also set an upper limit of 45 consecutive since n(n+1)/2 needs to be less than 1072
Let
n
sheets are missing starting from page number
x
. There will be
2
n
pages on the missing
n
sheets.
sum of page numbers is given to be 1072.
So,
x
+
(
x
+
1
)
+
(
x
+
2
)
+
(
x
+
3
)
+
⋯
+
'2n' times
=
1
0
7
2
x
+
(
x
+
1
)
+
(
x
+
2
)
+
(
x
+
3
)
+
⋯
+
(
x
+
2
n
−
1
)
=
1
0
7
2
2
n
x
+
(
1
+
2
+
3
+
⋯
+
2
n
−
1
)
=
1
0
7
2
2
n
x
+
2
(
2
n
−
1
)
(
2
n
−
1
+
1
)
=
1
0
7
2
2
n
x
+
(
2
n
−
1
)
×
n
=
1
0
7
2
n
[
2
x
+
2
n
−
1
]
=
1
0
7
2
=
1
6
×
6
7
n
[
2
(
x
+
n
)
−
1
]
=
1
6
×
6
7
since,
2
(
x
+
n
)
−
1
is a positive odd integer greater than 1 so it must equal to 67.
Hence, n=16.
Please let me know which pages were removed. (67-82) does not work
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pages (18-49)
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So the answer is 32? The question informed me that the answer is 16
A quick search on a 100-pages book using Mathematica gave tha right result: p a g e s 1 8 − 4 9
here is the code:
Select[Subsequences[Range@100],Total@#==1072&]