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

Modern Puzzles by Henry Ernest Dudeney: $180$

The Damaged Measure

Here is a new puzzle that is interesting, and it reminds one, though it is really very different,

of the classical puzzle by Bachet concerning the weight that was broken in pieces

which would then allow of any weight in pounds being determined from one pound up to the total weight of all the pieces.

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?

As an example, I give in the diagram the case of a $13$-inch rod with $4$ markings.

If I want to measure $4$ inches, I take $1$ and $3$; for $8$ inches, $6$ and $2$; for $10$ inches, $3$, $1$ and $6$; and so on.

Of course, the exact measure must be taken at once on the rod;

otherwise the single mark of $1$ inch repeated a sufficient number of times would measure any length, which would make the puzzle absurd.

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Measuring, Weighing, and Packing Puzzles: $180$. -- The Damaged Measure
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Combinatorial & Topological Problems: Miscellaneous Combinatorial Puzzles: $453$. The Damaged Measure