Solution:
1, 3, 9, 27
.
The trick with this one is realizing that “using the pieces
in tandem” means using the sum or
difference of two pieces to create measurements. For instance, if the ruler
were cut into segments that measured 1, 7, 13, and 19 inches, the 1‑inch
and 7‑inch segments could be placed end‑to‑end, and their sum would measure
8 inches, or the 13‑inch and 19‑inch segments could be placed side‑by‑side,
and their difference would measure 6 inches.
With the set given above, it’s possible to make the
following measurements:
1, 5, 6, 7, 8, 11,
12, 13, 14, 18, 19, 20, 21, 25, 26, 27, 31, 32, 33, 38, 39, 40.
Notice that the numbers occur in clusters. This happens
because of the 1‑inch segment. If the other pieces are used in tandem, the 1‑inch
piece can be used to make the number immediately above and below the result.
Also notice that some efficiency is lost because the difference between
consecutive pieces is always six: 19 – 13 = 6; 13 – 7 = 6; and 7 – 1 = 6.
These two observations can be helpful. The solution should
include a 1‑inch segment, and the segments should not have a common difference.
From there, we can “build” a set intuitively.
Start with a 1‑inch segment. If the next smallest segment is
2 inches, it will be possible to measure lengths of 2 and
3 inches, but if a 3‑inch segment is included instead, then it’s possible
to get measurements of 2, 3 and 4 inches, which is obviously better.
What if a 4‑inch segment were used instead of the 3‑inch segment? That would
allow for measurements of 3, 4, and 5 inches, but there would be no way to
get a measurement of 2 inches. There will be similar gap if a segment
larger than 4 inches is used. Consequently, the second smallest segment
should measure 3 inches.
With a 1‑inch and 3‑inch segment, measurements of 1, 2, 3,
and 4 inches are possible, which means that measurements for 4 inches
on either side of the two remaining numbers would be possible. A little thought
reveals that a 9‑inch segment would be a good choice; combined with the
measurements from the 1‑inch and 3‑inch segments, any measurement from 5
to 13 inches would then be possible. Choosing a segment less than
9 inches would allow for duplication of some measurements, and choosing a
segment greater than 9 inches would cause a gap.
Finally, the sum of the three segments chosen thus far is 1
+ 3 + 9 = 13. Consequently, the remaining segment must
be 40 – 13 = 27 inches. This works well, since any measurement from 1
to 13 inches is possible using the 1‑, 3‑, and 9‑inch segments, and
combining these measurements with 27 inches allows for any
measurement from 27 – 13 = 14 up to
27 + 13 = 40 inches.
