![]() And if they did, they might not make sense to them, anyway. Many grownups don’t know the actual definitions. So 14 can be written like 2 x 7, and 85 can be written as (2 x 42) + 1. In other words, an even number can be written like 2 x n, where n is some other number and an odd number is (2 x n) + 1, where n is another number. *Note: “number” here really means integer, but typically we just use whole numbers at the younger ages. ![]() Some Mathematical DefinitionsĪ number* is said to be even if it can be written as the product of another number and the number 2.Ī number is said to be odd if it can be written as the product of another number and 2, plus 1. We might as well say Bob numbers and Fred numbers. So even and odd are arbitrary names for alternating numbers. The words even and odd have no corresponding meaning in our standard English vocabulary for kids to anchor to.When you list even numbers, they alternate.Everyone over the age of 15 or so should have a general idea of which numbers are even and which numbers are odd.īut even-ness and odd-ness are NOT easy concepts. Even and odd numbers are considered a basic concept. Are the concepts of even and odd basic? Basic but Not Easy “I know, it’s a pretty basic concept, “ she told me, “but I’m just so excited he finally got it.” I was visiting with my sister yesterday and she was excited that her 8 year old son had finally grasped the idea. I didn't think it was easy to find a solid proof with either of them, so I'd be interested to hear from you if anyone makes any progess.Īnother obvious way of extending the problem is to try considering the number of ways of expressing a number as the sum of four odd numbers, or five or more.You know about even and odd numbers, right? You might find it easier to work with this version of the problem in your attempt to prove classĢYP's formula. The equation $a + b + c = n$ (for odd $a,b,c,n$) has as many solutions as this equation: $$(a+1) + (b+1) + (c+1) = (n+3).$$ Each of the bracketed quantities is even, so dividing through by a factor of 2 gives $$x + y + z = t $$ where $x,y,z$ are any integers, and $t$ is any integer greater then 2. If you thought that the pattern forming above looked suspiciously similar to that which we saw previously for only odd numbers, you would be correct, as I am about to demonstrate. The table above shows the number of ways of summing any 3 numbers to achieve the required total $t$. This will be the number of solutions to the problem because for each pair $\ Now all that remains is to count the number of possibilities for $a$ for each possible value of $c$. The smallest possible value of $a$ is $a= n-2c$ (which occurs when $n-2c> 0$ and $b=c$) and the largest is $a=(n-c)/2$ (which occurs when $a=b$). Once we've decided the range of possibilities for $c$, the possibilities for $a$ can also be limited. This restricts the range of values of $c$ to the interval $n/3 \leq c \leq n-2$. This is evident when you consider that $c$ is defined to be the largest of $a,b,c$ and that they must sum to $n$. Now that we've done that, we can also say that $c$ is at least the smallest odd integer greater than or equal to $n/3$ (where n is the required total). This way none of the solutions are repeated by having the same numbers in a different order. ![]() First of all, you should start with a trick which often comes in handy if you are trying to find a certain number of solutions and the order doesn't matter, that is to label them $a,b,c$ and define To do this you would need to consider how to limit your search. If you want to solve the problem differently, you might be interested in programming a computer to find the number of solutions for you. Will probably be much more difficult to show conclusively that their result concerning 3 odd numbers is correct. ![]() For each of the $k$ odd numbers there will be another such that the sum of the two is $n$ and the two cases occur according to whether $k$ is even or odd. If the even number $n$ is equal to $2k$, then the number of odd numbers less than $n$ is $k$. To start with, class 2YP found that $P_2(n) = n/4$ when $n$ is divisible by 4 and $P_2(n) = (n+2)/4$ when $n$ is an even number not divisible by 4. I'd like to introduce the following notation: let $P_x(n)$ be the number of ways in which $n$ can be expressed as the sum of $x$ odd numbers where we only count each set of $x$ numbers once, that is we ignore the order in which the numbers occur. They investigated the number of ways of expressing an integer as the sum of odd numbers. You may have seen the solution by Class 2YP from Madras College to a problem which they were inspired to consider after working on the problem called Score from the June Six. ![]()
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