MAA Blog: Playing with Place Value

By James Tanton @jamestanton, Facebook: James Tanton, LinkedIn: James Tanton

The invention of place value is one of humankind’s greatest intellectual achievements. Yet, as natural and straightforward as it seems to us today—that “333,” for instance, represents three-hundreds and three-tens and three-ones despite the repeated use of the one symbol “3”—is a concept that is subtle and nuanced. And the English language itself does not help matters. Although we’ll happily say twelve-thousand and two, and two-hundred and twelve, we wouldn’t dare say twelve-hundred and twelve-ty twelve. (If we did, what number are we describing?)

Place-value thinking permeates the entire mathematics curriculum, from early-grade base-ten arithmetic through to high-school and college polynomial, “base x,” arithmetic, and even to some work on infinite series. (In addition to this, one can explore fractional bases, negative bases, and the like, to swiftly enter the realm of unsolved mathematics.) It is often a delightful surprise to high-school students, and to many educators too, to see the same one storyline at the base of seemingly disparate arenas of mathematics.

Over 6.5 million students and teachers across the planet have been playing deeply with the power of place-value through the efforts of the Global Math Project and as the fifth-annual Global Math Week approaches—October 10 – 17 (a generous week!)—I’ve found myself musing on the interplay between base-x and base-ten thinking.

Might you and your students enjoy mulling on this small collection of place-value puzzlers that lie at this intersection?

Puzzle 1: The code of ones and zeros 110 represents a composite number no matter the base in which one interprets the code. Why? (Assume we are working with integer base values of at least two.)

110 = six in base two.

110 = twelve in base three.

110 = forty-two in base six

110 = one-hundred-thirty-two in base eleven.

Is there a code of ones and zeros that represents a prime number in each and every positive integer base?

Puzzle 2: You are told that a polynomial p , of unspecified degree, has all its non-zero coefficients positive single-digit whole numbers. (For example, p might be given by p(x)= 8x¹⁰ + x⁷ + 3x + 2.)Your job is to deduce what the polynomial must be from one piece of information. You may choose one integer input value n and be told the value of p(n). What value for n shall you choose?

Puzzle 3: Each non-zero coefficient of a polynomial p is either 1 or -1. Again you must deduce what p is from being told the output value p(n) from just one integer input value n. What value of n might you choose?

Puzzle 4: A polynomial p has coefficients from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5} . You do not know its degree. Explain why knowing the value of p(10) allows you to deduce what the polynomial is.

Puzzle 5: (This puzzle generalizes the previous three puzzles.)

You are told that each non-zero coefficient of a polynomial p is a positive or negative integer with absolute value no larger than a specific integer M. You may ask for the value p(n) for one integer input value n to determine what p must be. Give a value of n that would do the job.

Puzzle 6: (This puzzle is a classic.)

Each non-zero coefficient of a polynomial p is a positive integer. Your job is to deduce what the entire polynomial must be from just two pieces of information.

You may choose a number a and be told the value of p(a). Then, you may choose a second number b and be told the value of p(b).

What numbers a and b could you select?

BRIEF SOLUTIONS

Solving Puzzle 1: Suppose a code of 0s and 1s contains k 1s. If k=1, then the code is readily seen not to be prime in all bases. Otherwise, interpret the code as a number given in the specific base b=k+1. Now, the binomial theorem shows that b^r=(k+1)^r=k(something)+1 for each non-negative integer r. Thus, the code in base b is a number of the form

and so is a multiple of k and not prime.

Further: Prove that for a polynomial p with non-zero positive integer coefficients that, for each integer n, we have that p(p(n)+1) is a multiple of p(n). Prove that a polynomial with non-negative integer coefficients must have infinitely many outputs that are composite numbers for integer inputs.

Further Still: Show that every integer, positive or negative, can be written as a sum of distinct powers of -2, and that the sum is unique for each integer. (Welcome to base -2 using the digits 0 and 1!)

Still Further: The code 101111 gives the prime value 47 if interpreted as a base-two number, the prime value 283 if interpreted as a base-three number, the prime value 1109 if interpreted as a base-four number, but, alas, the value 3281 = 17 x 193 , not prime, if interpreted as a base-five number.

The code 10010111 is prime in bases two through five, and 111110100001 is prime in bases two through six. (See https://oeis.org/A086884 .)

For each positive integer N, is there sure to be a code of zeros and ones that represents a prime number in each of the bases two through N?

Solving Puzzle 2: A polynomial with non-negative integer coefficients can be interpreted as a number in base x with positive number “digits.” For example, p given by p(x)= 8x¹⁰ + x⁷ + 3x + 2 represents the “number” 8|0|0|1|0|0|0|0|0|3|2 in base x (and this is the base-ten number 80,010,000,032 if x=10).

If all the non-zero coefficients of p are single-digit whole numbers, then the value of p(10) completely determines the polynomial: the digits of this base-ten number match the coefficients of the polynomial.

(One can also deduce the polynomial from knowing the values of p(100) and p(1000), for instance.)

Further: Would knowing the value of p(0.1) allow you to deduce p? Would knowing the (full) value of p(π) do the trick too?

Solving Puzzle 3: Every integer has a unique representation in base three using the digits 0, 1, and -1.

(For a positive integer N, write it in base three using the digits 0, 1, and 2 as per the usual manner. Then, working right to left, replace each 2 with -1 and add one to the digit one place to its left and conduct any carries that may induce. If N is negative, do this for -N and then change the signs of the digits in the final representation. This establishes that the claimed representation exists. For uniqueness, start comparing the rightmost digits of two possible representations.)

To solve the puzzle, ask for the value of p(3) and write the answer in base three using the digits 0, 1, and -1. The digits you see will precisely match the coefficients of the polynomial.

Further: Is n=3 the only option for solving this puzzle?

Solving Puzzle 4: Every integer N has a unique base-ten representation with digits from the given set. (To see this, if N is positive and has a digit in its usual base-ten representation too large, add -10 to that digit and 1 to the digit one place to this left. Repeated application of this approach gives a representation in the desired form. Similar work shows that a negative integer can be so represented too. Uniqueness of representations again follows by comparing the rightmost digits of two possible representations.)

Writing the value of p(10) in base-ten with the digits {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5} reveals the polynomial p: the digits you see precisely match the coefficients of the polynomial.

Solving Puzzle 5: Let b=2M+2. Then, as per the method of the previous puzzle, every integer has a unique representation in base b using the “digits” {-M, -(M -1), …, -1, 0, 1, 2, …, M + 1}. Ask for the value of p(b) and write this value in base b using these digits. (One just shan’t happen to use the digit M+1.) The digits you see match precisely the coefficients of p.

Solving Puzzle 6: Here p is a polynomial with non-negative integer coefficients. First ask for the value p(1). This gives you the sum of all those coefficients. Let b=p(1)+1. Then each coefficient of p is smaller than b, which means that the number p(b), when written as a number in base b, has each “digit” in the range 0, 1, …, b-1 and these digits precisely match the coefficients of the polynomial. So ask for the value of p(b) next, write that value in base b, and see what those coefficients are.

Further: Our solution here fails if we are required to choose numbers a and b simultaneously: our choice of b depends on what we hear as the result of first choosing a=1. Do you think this puzzle can be solved if you must submit two chosen inputs at the same time? Would being allowed to submit three inputs simultaneously help matters? Seven? Seven-hundred?

To learn more about the Global Math Project and its approach to presenting the story of place-value to the world, see https://globalmathproject.org/exploding-dots/.

James Tanton is the Mathematician-at-Large for the Mathematical Association of America, and Co-Founder of the Global Math Project.