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

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

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November 13, 2007 at 4:17 am |

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).

November 13, 2007 at 7:27 am |

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

November 13, 2007 at 7:53 am |

The intent was that “6-digit number” means a number with 6 digits, the first of which is

not0. Although, it might be easier to solve if we proceed inductively and then subtract the “shorter” numbers as necessary.