Henry Ernest Dudeney/Modern Puzzles/180 - The Damaged Measure/Solution

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Modern Puzzles by Henry Ernest Dudeney: $180$

The Damaged Measure
A young man has a yardstick from which $3$ inches have been broken off,
so that it is only $33$ inches in length.
Some of the graduation marks are also obliterated, so that only eight of these marks are legible;
yet he is able to measure any given number of inches from $1$ inch up to $33$ inches.
Where are these marks placed?


Solution

Divide the $33$-inch measure into the following sections:

$1$, $3$, $1$, $9$, $2$, $7$, $2$, $6$, $2$

Another solution is:

$1$, $1$, $1$, $1$, $6$, $6$, $6$, $6$, $5$

According to Dudeney himself, there are $16$ different ways of doing it.


Historical Note

Martin Gardner reported in $1968$ that no general rule had yet been found for the ruler problem.

However, considerable progress had been made since Dudeney first proposed it.

In particular, John Leech discovered in $1956$ that eight marks are sufficient to mark a $36$-inch ruler so that all integer lengths from $1$ to $36$ can be measured, outdoing Dudeney's solution by $3$ inches.


Sources

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