Olivia Morrison
Prof. Winkler
Writing Assignment #2
04.25.2013
Counting on Fingers: A New Perspective
Martin Gardner’s “Finger Count” puzzle, found in his The Colossal Book of Short
Puzzles and Problems, involves strategy and new perspectives – two hallmarks of a great
puzzle. The deceptively simple solution to this particular puzzle arises only through a
specific strategy that would not likely have been the reader’s first thought in attempting
to find an answer. As such, the puzzle forces the reader to look at the problem in a new
light in order to reach the correct solution. This forced new perspective, together with the
eventual simplicity of the solution, makes this a genuinely good puzzle in terms of
structure and strategy.
Gardner’s “Finger Count” puzzle is as follows:
On last New Year’s Day, a mathematician was puzzled by the
strange way in which his small daughter began to count on the fingers of
her left hand. She started by calling the thumb 1, the first finger 2, middle
finger 3, ring finger 4, little finger 5, then she reversed direction, calling
the ring finger 6, middle finger 7, first finger 8, thumb 9, then back to the
first finger for 10, middle finger for 11, and so on. She continued to count
back and forth in this peculiar manner until she reached a count of 20 on
her ring finger.
“What in the world are you doing?” her father asked.
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The girl stamped her foot. “Now you’ve made me forget where I
was. I’ll have to start all over again. I’m counting up to 1962 to see what
finger I’ll end on.”
The mathematician closed his eyes while he made a simple mental
calculation. “You’ll end on your -------,” he said.
When the girl finished her count and found that her father was
right, she was so impressed by the predictive power of mathematics that
she decided to work twice as hard on her arithmetic lessons. How did the
father arrive at his prediction and what finger did he predict? (Gardner 6364)
Before looking at the solution to this puzzle, we can analyze the puzzle itself in
terms of a few key factors. This is a visual puzzle – the reader can look at his or her own
hand to better understand the counting style of the mathematician’s daughter. If the
reader really wanted to, he or she could, in fact, count all the way to 1962 to arrive at the
answer. This endeavor would require a long time and a large amount of concentration,
however, and so there must be a simpler way to reach the solution. This new pathway to a
solution requires looking at the problem in a new light. Instead of simply relying on the
first instinct of counting, the reader must think of a new, more strategic and direct way to
solve the puzzle. In addition, the puzzle asks the reader what strategy the mathematician
used to arrive at the identity of the final finger. So, the reader must also reach his or her
conclusion with the mathematician’s strategy in mind; it is not enough to find just the
finger on which the mathematician’s daughter would have ended when she reached 1962.
While this is not a paradoxical problem – the answer is clearly one of five fingers, and is
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definite – it does force the reader to consider mathematical possibilities of reaching the
answer in a way other than simply counting fingers until arriving at the desired 1962.
Thus, the puzzle’s main attribute is its forcing of the reader to use an innovative and
strategic approach to finding a solution.
The solution to this puzzle is clear, once it is revealed. It involves simple math,
instead of the first instinct of counting:
When the mathematician’s little girl counted to 1,962 on her fingers,
counting back and forth in the manner described, the count ended on her
index finger. The fingers are counted in repetitions of a cycle of eight
counts…It is a simple matter to apply the concept of numerical
congruence, modulo 8, to calculate where the count will fall for any given
number. We have only to divide the number by 8, note the remainder, then
check to see which finger is so labeled. The number 1,962 divided by 8
has a remainder of two, so the count falls on the index finger.
In mentally dividing 1,962 by 8 the mathematician recalled the rule that
any number is evenly divisible by 8 if its last three digits are evenly
divisible by 8, so he had only to divide 962 by 8 to determine the
remainder. (76)
The specific strategy embedded in this solution is not necessarily readily apparent
to the reader. The cycles of 8 counts are a crucial factor, and once the reader is able to
recognize that specific pattern, it becomes much easier to reach the overall solution to the
puzzle. This easy-once-reached solution makes for a good puzzle: it is a challenge to find
the answer, but once the puzzle is solved, it seems much simpler than it originally did.
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The solution does not require knowledge of advanced math, and is thus appealing to the
average puzzle solver. Both the puzzle and solution are written in plain, easy-tounderstand terms, and neither tries to trick or confuse the reader. He or she is given a fair
starting point – absent any trickery or deception – from which to attempt solving the
puzzle. The solution is also quite easy to follow, should the reader wish to trace it back
and better understand the problem.
Beyond the format of the puzzle and solution, the answer also makes sense. The
cycle of eight is easily recognizable (or at least, can be brought about through counting
on one’s fingers in the same manner as the mathematician’s daughter a few times). Once
the reader comes across this cycle, the final answer is not far off. If he or she is able to
identify the manner in which to utilize the newfound pattern, he or she will solve the
puzzle. Dividing the total number, 1,962, by 8 will give a remainder that can then be
counted off on one’s fingers to reach the final digit of the overall count – in this case, the
index finger. It is the eight-count cycle that cracks this puzzle open. Without it, the reader
must resort to counting to 1,962 on five fingers. Using the cycle of eight, the reader is
presented with a much faster avenue through which he or she can arrive at the solution to
the puzzle.
Gardner’s “Finger Count” puzzle is exemplary in its presentation of a seemingly
complicated puzzle with a simple solution. There is one specific “key” to the solution: the
eight-count cycle. Once the reader picks up on this cycle, the solution becomes
abundantly clear. This problem is a visual one: the reader can count on his or her own
fingers to help him or her arrive at the solution, thereby making the problem more
tangible than abstract. Perhaps most importantly, the “Finger Count” puzzle forces upon
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the reader a new perspective and strategy necessary in order to find the correct solution.
Because Gardner specifically asks for the mathematician’s strategy in reaching the
answer to his daughter’s question, the reader is forced beyond the instinctual counting
and into a new way of thinking – one involving math and patterns and, eventually, an
eight-count cycle.
Works Cited
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Gardner, Martin. The Colossal Book of Short Puzzles and Problems: Combinatorics,
Probability, Algebra, Geometry, Topology, Chess, Logic, Cryptarithms,
Wordplay, Physics, and Other Topics of Recreational Mathematics. Ed. Dana
Richards. New York: Norton, 2006. Print.
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