Using Math Rules: An Example

I can't commend Sergey Dorichenko's book strongly enough. This is a collection of week-by-week sets of problems for a math circle attended by both young and more experienced students so that the problem range from simple to more sophisticated. But even most simple problem afford a glimpse of real mathematics far removed from the commonly held views.

Which is greater, $333333\times 444444$ or $222222\times 666667,$ and by how much? (Problem 1.2)

The problem has an appeal in that it may be offered to children yet unused of handling 6-digit numbers who may still be able to absorb the solution without regard to the number of digits involved. Also, the solution - however you approach it - appears to reference the three basic properties of arithmetic operations: commutativity, associativity, and the distributive law. These do not have to be mentioned by name, but the solution makes it rather necessary to point out the properties they signify. Moreover, knowing these rules permits to answer and generalize the problem without carrying out a single calculation.

Commutativity: $333333\times 444444 = 444444\times 333333.$ Associativity:

$444444\times 333333=222222\times 2\times 333333=222222\times 666666.$

Distributivity:

$\begin{align} 222222\times 666667 &= 222222\times (666666+1)\\ &=222222\times 666666 + 222222\times 1\\ &=222222\times 666666 + 222222. \end{align}$

Finally, $222222\times 666667 = 333333\times 444444 + 222222,$ meaning that $222222\times 666667$ is bigger than $333333\times 444444$ by $222222.$

The decision to handle 6-digit numbers is probably intended to suggest that direct calculation of the given products might not be such a great idea, thus channeling the attention to something more sophisticated.

The problem allows for multiple variations and also could be posed with shorter or longer numbers. Here's one of the less trivial modifications:

Which is greater, $3333\times 4445$ or $2222\times 6667,$ and by how much?

$\begin{align} 3333\times 4445 &= 3333\times 4444 + 3333\\ &=6666\times 2222 + 3333\\ &=6667\times 2222 -2222 + 3333\\ &=6667\times 2222 + 1111. \end{align}$

Reference

  1. Sergey Dorichenko, A Moscow Math Circle: Week-by-Week Problem Sets (MSRI Mathematical Circles Library, 2012)

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