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Here are a couple problems to work on during your lunch break:

1. Let a and b be real numbers such that a+b=1. What is the relationship between a2+b and a+b2? Does this change if we allow a and b to be complex numbers?
2. Let n be a 1-digit whole number. How many 6-digit numbers contain at least one n? Generalize this to k-digit numbers.

3 Responses to “Your Brain Games for Today”

1. TwoPi Says:

1. Does this change if we allow a and b to be polynomials? Does this change if we allow a and b to be square matrices?

2. The answer depends (slightly) on n (unless “6-digit number” includes all the numbers from 0 to 999999). Generalizing to k-digit numbers is fairly straightforward (if 6 is replaced by k) or rather more interesting (if 1 is replaced by k).

2. Ξ Says:

I think #1 is really weird. I get it, but I still think it’s really weird. Man, I love math!

3. Batman Says:

The intent was that “6-digit number” means a number with 6 digits, the first of which is not 0. Although, it might be easier to solve if we proceed inductively and then subtract the “shorter” numbers as necessary.