Problem of the Week Problem D and Solution The Clock Works
Problem
Halina’s clock uses a digital LED display where each digit is
represented by seven LED segments that are either on or off, as
shown.
Seven segments of equal length are arranged to form a figure of two
identical squares with one on top of the other sharing a common side.
Ten such figures are shown side by side, each having some of the
segments highlighted to represent the ten digits \(0\) - \(9\). The list of highlighted segments (or
sides of the two squares) for each digit is as follows.
Digit \(0\): All segments except
for the common side of the two squares.
Digit \(1\): Right side of both
squares.
Digit \(2\): Top, right, and
bottom sides of the top square; left and bottom sides of the bottom
square.
Digit \(3\): Top, right, and
bottom sides of the top square; right and bottom sides of the bottom
square.
Digit \(4\): Left, bottom, and
right sides of the top square; right side of the bottom square.
Digit \(5\): Top, left, and
bottom sides of the top square; right and bottom sides of the bottom
square.
Digit \(6\): Top, left, and
bottom sides of the top square; all sides of the bottom square.
Digit \(7\): Top and right sides
of the top square; right side of the bottom square.
Digit \(8\): All
segments.
Digit \(9\): All segments except
for the left side of the bottom square.
Sometimes some of the LED segments stop working. When the topmost
horizontal LED segment stopped working, both the digit \(1\) and the digit \(7\) appeared as shown.
This was a problem because Halina couldn’t distinguish between
them.
Halina replaced the broken LED segment, but then a week later found
that a different LED segment had stopped working. However, this time,
she was still able to distinguish between all ten digits.
What is the largest number of LED segments that can be broken at the
same time, while still allowing Halina to distinguish between all ten
digits?
First we will label the seven LED segments as shown.
If segment \(A\) is broken, Halina
cannot distinguish between digits \(1\)
and \(7\).
If segment \(B\) is broken, Halina
cannot distinguish between digits \(6\)
and \(8\).
If segment \(E\) is broken, Halina
cannot distinguish between digits \(5\)
and \(6\), and also between digits
\(8\) and \(9\).
If segment \(F\) is broken, Halina
cannot distinguish between digits \(3\)
and \(9\).
If segment \(G\) is broken, Halina
cannot distinguish between digits \(0\)
and \(8\).
If both segments \(C\) and \(D\) are broken, the ten digits would appear
as shown.
Ten figures of seven segments forming two squares sharing a common
side. The list of highlighted segments (or sides of the two squares) for
each digit is as follows.
Digit \(0\): Left, top, and
right sides of the top square; left side of bottom the square.
Digit \(1\): Right side of top
square.
Digit \(2\): Top, right, and
bottom sides of the top square; left side of the bottom square.
Digit \(3\): Top, right, and
bottom sides of the top square.
Digit \(4\): Left, bottom, and
right sides of the top square.
Digit \(5\): Top, left, and
bottom sides of the top square.
Digit \(6\): Top, left, and
bottom sides of the top square; left side of the bottom square.
Digit \(7\): Top and right sides
of the top square.
Digit \(8\): All sides of the
top square; left side of bottom square.
Digit \(9\): All sides of the
top square.
Since each of these digits are unique, Halina can distinguish between
them. Therefore, at most \(2\) LED
segments can be broken at the same time.
