Los Angeles Math Circle

3/1/2020 -- Beginners 2A: Polyhedra I Continued

Warm-Up: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090.

The insight: if you rewrite 998 as (1000-2) and 992 as (1000-8), multiply the two we get 1000*1000 - 2*1000 - 8*1000 + 8*2. Hence we get (1000-2-8)*1000 + 16 = 990,016.

We will be continuing our topic on Polyhedra from last week!