James Tanton

The equation of a circle of radius 10 is x^2 + y^2 = 100. Replace each variable with its "floor." What does the graph of (floor(x))^2 + (floor(y))^2 = 100 look like? [Here: Floor(a) = the largest whole number less than or equal to a. E. g. Floor(3.1)=3, Floor(-3.1)=-4.]

The floor function rounds a real number down to the nearest whole number. E.g. Floor(3.1)=3, Floor(-3.1)=-4, Floor(3)=3. The grapy of y=(Floor(x))^2 is below. What do the graphs of these variations look like? y=x*Floor(x) Floor(y)=x^2 Floor(y)=x*Floor(x) y=Floor(x^2)

Floor(x) = the largest integer less than or equal to x. e.g. Floor(3.1) = 3, Floor(3) = 3, Floor(2.9) = 2, Floor(-3.1) = -4. Desmos can draw the graph of y = x^2, but not its chunky counterpart Floor(y)=(Floor(x))^2. What should the chunky graph look like?

Three points (a,a^3), (b,b^3), (c,c^3) on the graph of y = x^3. Where does the quadratic that passes through those points cross the y-axis?

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In my atrocious handwriting, here's an argument showing where a line crosses the curve y = x^3 a third time. Can anything reasonable be said about the same setup for y = x^5? If a line crosses its graph at (a,a^5) and (b,b^5), where does it cross a third time?

Where does the line connecting (a,a^3) and (b, b^3) again intercept the curve y = x^3 ?

A polynomial P with integer coefficients has P(a)=b, P(b)=a and P(c)=d, P(d)=c for four distinct integers a, b, c, and d. What must be true about those four integers?

More Polynomial Fun: Is there a polynomial P with integer coefficients with P(3)=17, P(17) = 3 and P(8)=12, P(12)=8? How about one with P(3)=17, P(17) = 3 and P(9)=12, P(12)=9?

@_Manuel_Ruiz_ Oops! Yes indeed!

@jamestanton I think the question is meant to be about a degree n-1 polynomial. Otherwise, we have an infinite amount of solutions, and one of them will have P(n+1) = 1/(n+1).

A degree n polynomial P wants to be the expression 1/x. It has P(1)=1, P(2)=1/2, P(3)=1/3, ..., P(n)=1/n. How close is the value of P(n+1) to being 1/(n+1)?

OOPS! I meant degree n-1! Here's the correct puzzle: A degree n-1 polynomial P wants to be the expression 1/x. It has P(1)=1, P(2)=1/2, P(3)=1/3, ..., P(n)=1/n. How close is the value of P(n+1) to being 1/(n+1)?

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A fourth-degree polynomial P wants to be the rational expression 1/x. It has P(1)=1, P(2)=1/2, P(3)=1/3, P(4)=1/4, and P(5)=1/5. What is the value of P(6)?

A polynomial P has integer coefficients. If P(-1), P(0), and P(1) are each multiples of three, must P(9437) also be a multiple of three?

There is no polynomial with integer coefficients along with three distinct integers a, b, c such that P(a) = b, P(b) = c, and P(c) = a. What about a polynomial P with real coefficients? Might one be able to find three real numbers a,b,c so that P(a) = b, P(b) = c, and P(c) = a.

@_Manuel_Ruiz_ Indeed -- and this is the next puzzle. Can you establish that the coefficients here can't all be integers?

@jamestanton That works for non-integer coefficients.

The polynomial P(x) = a + b - x has P(a) =b and P(b) = a. Find a polynomial with integer coefficients along with three distinct integers a, b, c such that P(a) = b, P(b) = c, and P(c) = a.

A polynomial P with integer coefficients has P(a) = 3, P(b)=2, and P(c) = 1 for three integer inputs a, b, and c. Explain why a, b, and c must be consecutive integers.