kitchen table math, the sequel: arithmetic versus algebra

## Sunday, October 11, 2009

### arithmetic versus algebra

Consider the set: {1, 3, 5}
:: If elements of this set can be selectively added together to yield some number q, what is its maximum?
:: Are there any odd values between 1 and qmax that q cannot hold?
:: Are there any even values between 1 and qmax that q cannot hold?

Consider the set: {1 , 3 , 5 , 7 , 9 , 11}
:: What is the maximum of q?
:: What odd values between 1 and qmax can q not hold?
:: What even values between 1 and qmax can q not hold?

Consider the set: {1 , 3 , 5 , 7 , 9 , 11, 13}
:: What is the maximum of q?
:: What odd values between 1 and qmax can q not hold?
:: What even values between 1 and qmax can q not hold?

Consider a finite set of odd numbers, {1, 3, 5, 7 ... n}
:: What is the maximum of q, in terms of n?
:: Find the odd values between 1 and qmax that q cannot hold, in terms of n.
:: Find the even values between 1 and qmax that q cannot hold, in terms of n.

Consider the set: {-6, -4, -2, 1, 3, 5, 7}
:: What is the maximum and minimum of q?
:: What odd values between qmin and qmax can q not hold?
:: What even values between qmin and qmax can q not hold?

Consider a finite set: {-2n, -2(n-1), ... -4, -2, 1, 3, 5 ... 2(n-1)+1, 2n+1 }
:: What is the maximum and minimum of q?
:: What odd values between qmin and qmax can q not hold, in terms of n?
:: What even values between qmin and qmax can q not hold, in terms of n?

Lsquared said...

Just making sure I understand the problem. For the first set: {1, 3, 5}, qmax would be 9, correct? "Selectively added together" isn't something I'm sure I understand.

Yeah, that is, 1 can be added to 3, 3 added to 5, 1 added to 5, or everything can be added. But each element can only be added once (you can't do 5 + 5 + 5 for example).

Technically qmin here is 0 (null sum) which is why I specified 1 as a lower bound, lol.