He asks, “If x and y are counting numbers (whole numbers above zero), what numbers do the terms (x÷2), (3÷y), and (y-x) represent?”
The answer Marilyn gives is completely unsatisfactory.
X and Y have to be counting numbers, but NOWHERE does the question state that (x÷2), (3÷y), or (y-x) have to be counting numbers. Marilyn’s answer implies that they do. X could be 2,374,893, and Y could be 45,349,249,732,901,245,647. You still get perfectly valid numbers if you plug those variables into the equations.
As long as it’s stipulated that the answers to the formulas are also counting numbers, it’s simple:
If (x÷2) is a counting number, then x has to be even.
If (3÷y) is a counting number, y can only be 1 or 3.
If (y-x) is a counting number, then x must be less than y.
There is no even number less than 1, so y must equal 3 and x is therefore 2, and the answer to all three formulas is 1. The only problem is that the puzzle as it appeared in MVS’s column omitted that crucial stipulation.
EDIT: I mean I don’t understand how I came up with the correct answer when given an incomplete question. My only conclusion is that the question is way easier than the writer intended.
The question isn’t clear. But if you assume that all of the numbers involved have to be counting numbers, x must be an even number, y must be a multiple of three that is not larger than three, and y must be larger than x. The solution is x=2 and y=3.
But then, her assertion that it is a “dandy” logic puzzle is false. It’s not even an acceptable logic puzzle (acceptable logic puzzles do not permit the solver to make any assumptions at all, but require that only the information given in the setup be used).
Maybe. I once ran across a Sudoku that could not be solved unless you assumed that the answer was unique. I had narrowed a particular box to two possibilities; one of those possibilities left other squares in an ambiguous state, so I knew the other move had to be correct.
Uniqueness is guaranteed in a properly-constructed Sudoku, but is it a fair assumption in other cases? I don’t know. I could say: “I’m thinking of a counting number between 0 and N (exclusive). What is it?” If you assume there is a unique answer, then N must be 2 (or a real greater than 1), and the number I’m thinking of must be 1. That seems a little weird but isn’t entirely unfair.
At any rate, the question from the column was certainly bad, since even assuming a unique answer doesn’t tell you which other assumptions you must make. Assuming counting numbers is certainly a good guess but not the only possibility.
Kinda reminds me of an old puzzle requiring you to figure out the ages of a person’s children. It’s ambiguous after the first two statements, but only coalesces after the third, in which the parent says something that seems initially irrelevant like “My oldest daughter has blue eyes”, which eliminates possibilities in which the older children are twins or something (though it doesn’t really, since even among twins, one is delivered first and could be considered “older”).
I agree that it’s not a very good puzzle but not for your reasons. The only difficulty involved in this puzzle was figuring out what was being asked. Once you figured that out, the math was trivially easy. If the puzzle had been better constructed, it would have been worthless as a challenge.
If that was the only way to solve the puzzle, then you had a puzzle with (at least) three valid solutions, which is not only an improperly-designed Sudoku, but also meant that you used an incorrect assumption to solve it. If the puzzle did not in fact have multiple solutions, then that means that there must have been some other method which could have been used to solve it, just one that you either didn’t know or lacked the brainpower to apply it.
And Marilyn’s puzzle depends not only on assuming that the results of all calculations are positive integers; it also assumes something about the operations. One could say, for instance, that x = 7, y = 10, x÷2 = 3, 3÷y = 0, and y-x = 3 (which is what a computer would get from integer arithmetic on those numbers).
EDIT: And there’s another, much worse, problem with the puzzle and Marilyn’s solution: She starts off by saying that it involves “no math, only logic”. Math is logic, and she’s doing STEM education a grave disservice by perpetuating the misconception that it’s not.
It’s possible. I certainly have no doubt that if I fed the possibilities into a recursive computer solver, that two would have been rejected. Or even that if I myself had gone forward with the other solutions, I’d eventually found a contradiction. But another “rule” of Sudoku is that they only require “human logic,” and not deeply nested guess and check. There should always be a way forward based solely on the immediate board state.
I’ve built heuristic solvers that can solve any conventional Sudoku that I throw at it, so I’m familiar with the techniques (or at least I was…). It didn’t seem possible for this one, though. At any rate, the technique is valid for Sudoku though I don’t know if it was intentional here.
I’ve never seen a strong distinction between “human logic” and “deep guess and check”. Most of the “human methods” are just a special case of guess and check that’s become familiar enough that you can recognize it quickly, and different people will have different standards on how deep is “deep”.
I would disagree. There are puzzles I solve using logic not guessing and checking. I start with the information given to me and use it to place answers. I’m not guessing because I know what these answers must be based on the information I have. And these answers give me more information which I use to place more answers and so until I’ve completed the puzzle. No guesswork is involved.
Yeah, I disagree as well. A recursive solver is absolutely brain-dead simple and will solve any Sudoku. But it doesn’t deduce anything at all; it just tries possibilities until it finds something that doesn’t hit a contradiction. A “human” heuristic looks more like “Suppose you have N cells in a group that cover exactly N possible numbers. Then, you can reject all of those numbers from cells not in the subset.” There’s no guess and check here; it’s just a logical deduction from the way the rules are set up.
That’s not to say it doesn’t get a little fuzzy on the boundaries, but in practice there’s usually a clear distinction.
Of course there’s guess and check. The simplest “human” method of all consists of seeing that all but one of the numbers are eliminated. So you’re going “Hmm, is it a 1? No, there’s another 1 in that row already, so that’s a contradiction. Is it a 2? No, there’s another 2 in its 3x3 box, so that’s a contradiction”, and so on, until you find one number that isn’t a contradiction. What you’re doing there is guess-and-check, which is a perfectly valid logical method.
It maybe doesn’t feel like guess-and-check, because you’re not writing anything down until you know what the cell is for sure. But how deep you can go before writing anything down just depends on how good your memory is.
No, because the column appeared in a Sunday supplement, not Scientific American, and “counting numbers” is a term that means exactly the same thing, and does not need to be separately defined for the benefit of the pedestrian comprehension level of the readers. Pedantry will get you nowhere.