. In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers,

113; 137; 149; 155; 173;

accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?

_113_137_149_155_173_
the 6 spaces represent the possible locations for the missed number
and there are 5 possible numbers that could be the one that is wrong.
Of all of these 30 possible combinations, at least one of the
differences between consequtive numbers is preserved.  So the list of
possible common differences for the sequence are:
137-113=24
149-137=12
155-149=6
173-155=18
now since only one of the original numbers is wrong, that means either
113 or 137 is in the actual sequence.  Looking at each of these in
combination with the 4 possible common sequnces we get the following
8 sequences:
{first number, common difference} {sequence}
s1: {113,6} {113,119,125,131,137,143,149,155,161,167,173,179}
s2: {113,12} {113,125,137,149,161,173,185}
s3: {113,18} {113,131,149,167,185}
s4: {113,24} {113,137,161,185}
s5: {137,6} {107,113,119,125,131,137,143,149,155,161,167,173,179}
s6: {137,12} {101,113,125,137,149,161,173,185}
s7: {137,18} {101,119,137,155,173,191}
s8: {137,24} {89,113,137,161,185}

we are looking for a sequence of 6 consecutive values which contains
exactly 4 of the 5 original numbers and only introduces 2 new numbers
(one missed and one corrected).

The only sequences for which this work is s2 and s7 which both give the
sequence {113,125,137,149,161,173}

Posted by Daniel on 2013-09-27 13:29:00