Find the missing number:

series/paper.sizes New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Find the missing number:

37, 52, 74, 105, 148, 210, ?, 420, 594, 841, 1189

According to the question, this is not some obscure 'Mensa' type
problem, but is a sequence used in a very real and practical way.  It
seems that the missing number ought to be 297, but I can't find any
real justification for this.

-- Robert Alan Watts <robert@robwatts.demon.co.uk>, July 1995

Show solution

The "A" sizes of paper are defined as follows.  A0 is a sqrt(2)*1
rectangle with an area of 1 square metre.  To the nearest millimetre,
it is thus 1189x841 mm, and those numbers are the last two in the
sequence.

Then A$n$ is a rectangle whose long side is as long as A$n-1$'s short
side, and whose short side is half as long as A$n-1$'s long side.  (The
$'s are merely a clumsy way of indicating that they enclose a
mathematical expression.)  This gives rise to a halving sequence
starting with sqrt(sqrt(2)) m.  Convert to mm and round, take the
sequence to 11 terms, then reverse it, and we have the original
sequence.  The term 297 to be supplied is the height in mm of an A4
sheet of paper.

-- Richard Sabey <rsa@swindon.gpsemi.com>, July 1995