How many whole numbers can you count on your fingers?
Define "counting up to a number " as a whole number that you can represent in a restricted system while also being able to represent each of the whole numbers from to within this system.
For example, using the fingers of both of your hands, you can count up to -- from through and all of the whole numbers in between.
Given this definition, what is the highest number you can count up to using only the five fingers of your left hand ? (You can ignore "half a finger" or the creases in your finger; only entirely open fingers and closed fingers are allowed to represent numbers.)
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
This problem demonstrates the superiority of the binary representation of numbers over others when the only allowed placeholders are 0 (a closed finger) and 1 (an open finger).
The simplest, but the least useful, representation is to count from 0 through 5 using each of the fingers of your left hand (or right hand, doesn't matter). With all five fingers open, the highest number you can count up to in this system is 1 + 1 + 1 + 1 + 1 = 5 .
A slightly better representation is to start counting with the index finger and proceed towards the pinky finger. With all four fingers open, you can represent the number 4 . But now open your thumb and close all of the other four fingers. An open thumb represents the number 5 . With the thumb now open, again start counting (opening) with the index finger and proceed towards the pinker finger. With all five fingers open, the highest number you can count up to in this system is 5 + 1 + 1 + 1 + 1 = 9 .
An even better representation is to start counting with the middle finger and proceed towards the pinky finger. With all three fingers open, you can represent the number 3 . But now open your index finger and close all others. An open index finger represents the number 4 . With the index finger now open, again start counting with the middle finger and proceed towards the pinky finger. With all four fingers open, you can represent the number 7 . Now open your thumb and close all others. An open thumb represents the number 8 . With the thumb now open, again start counting with the middle finger and proceed towards the pinky finger. These four open fingers represent 8 + 1 + 1 + 1 = 1 1 . Now close the middle, ring, and pinky fingers and open the thumb and index finger only. This represents 8 + 4 = 1 2 . With thumb and index open, start counting again from the middle towards the pinky finger. With all five fingers open, the highest number you can count up to in this system is 8 + 4 + 1 + 1 + 1 = 1 5 .
Similarly, you can fill in the answer for starting counting with the ring finger and proceeding towards the pinky finger. Here, m i d d l e f i n g e r = 3 , i n d e x f i n g e r = 6 , t h u m b = 1 2 . With all five fingers open, the highest number you can count up to in this system is 1 2 + 6 + 3 + 1 + 1 = 2 3 .
Now, the most efficient: the binary. Pinky, as in all of the above systems, represents 1 . r i n g f i n g e r = 2 , m i d d l e f i n g e r = 4 , i n d e x f i n g e r = 8 , t h u m b = 1 6 . With all five fingers open, the highest number you can count up to in this system is 1 6 + 8 + 4 + 2 + 1 = 3 1 .
Using the same number of fingers, you can count up to a higher number in binary than in any other method (but only when the allowable placeholders are 0 (a closed finger) and 1 (an open finger)).