Last week’s Micro Softy dealt required a bit of algebra. This week’s puzzle requires knowledge of elementary calculus.

If this is too much math, come back next week for a puzzle about birthdays. For those who know enough calculus to be dangerous, this is a fascinating puzzle — one that usually stumps even my best graduate students.

We know that

(d/dn) n² = 2n.

But we also know

n² = n+n+n+…+n.

where there are n terms. Thus

(d/dn) n² = (d/dn) [n+n+n+…+n]

= 1+1+1+…+1 = n.

This is a different answer!

This week’s Micro Softy challenge: Resolve this inconsistency and rescue the honor of calculus!

Solution to Micro Softy 31: Elementary Algebra

Last week’s Micro Softy was short and sweet. The puzzle is to simplify the following function of x as much as possible.

(x-a)(x-b)(x-c)…(x-y)(x-z)

This expression can be expanded a little bit by explicitly adding one more term, (x-x),  near the end.

 (x-a)(x-b)(x-c)…(x-x)(x-y)(x-z)

This addition reveals the solution.

Since x-x=0 and anything multiplied by zero is zero, the expression simplifies to

(x-a)(x-b)(x-c)…(x-y)(x-z) = 0.

That is pretty simple.

(The Micro Softy is a weekly feature of Mind Matters News. Links to all the previous ones are available through this one.)

Monday Micro Softy 31: Elementary algebra. How much can an equation be simplified? In last week’s puzzle, the car was handling like a cement mixer because something was wrong with the tires. The solution lies in finding out what that is.

Monday Micro Softy 30: Driving a cement mixer. Why did the car with leaky tires suddenly start handling like a cement mixer, and then, just as suddenly, handle normally again? Micro Softy 29 shows that unambiguous facts can be stated in confusing language but cold hard logic can solve the puzzle.

Links to all the previous Monday Micro Softies

Monday Micro Softy 29: A funeral lament in four lines. The funeral director was puzzled by Dan’s description of his relationship to the deceased but there was no question that his grief was sincere Here, you will also find links to Microsofties 22 through 29 as well.

Monday Micro Softy 21: Finding More of the Deadly Fentanyl Pills. There, you will also find links to Microsofties 11 through 20 as well.

At Monday Micro Softy 11: What Happened to That Other Dollar?, you will find links to the first ten Micro Softies. Have fun!