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Which books contain practice questions — preferably with full solutions — to assist 15 year olds with the Fence Post or Off by One error?
Most students at my institution have not heard of this name, though some recognized the error. Some of them are hankering after more practice exercises, to assist them with preventing it! Even after I cover the concept in general, though without exercises, too many of them fail to spot and avoid it on exams and tests.
You only really need to give them one type of question:
Count the numbers in this list: [insert any finite easily indexed sequence]
Examples:
$4,5,6, \cdots, 27$
$112, 115, 118, \cdots, 334$
$4, 2.5, 1, \cdots, -29$
$4, 8, 16, \cdots, 2048$
$9, 16, 25, 36, \cdots, 400$
And then teach them to solve ALL of these problems by method of bijections, forming one-to-one pairings of lists by explicit rules.
For instance, take the second one: $112, 115, 118, \cdots, 334$
Subtract $100$ to everything in the list: $12, 15, 18, \cdots, 234$
Divide by $3$: $4, 5, 6, \cdots, 78$
Subtract $3$: $1,2,3, \cdots, 75$
So the answer is $75$.
There are infinitely many ways to do this problem, and it doesn't matter how many bijections you make, you will always end up at the correct answer. This is superior to the formula for the number of items in a sequence for several reasons:
It is critically important to teach this method for some major reasons. Firstly, bijections are the foundation of combinatorics. It is how we define counting in addition to being an exceedingly powerful abstract problem solving tool. Secondly, this method gives students a way to guarantee they have the correct answer; it is the way in which we check our answer to counting problems in the same way that plugging values into an equation is how we check our answer in algebra and how proofs are how we check our answer in geometry. And finally, if they get used to applying this method and thinking about counting in this way, it builds excellent habits and avoids many common pitfalls.
