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Subtraction and division cages
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Author:  rhpzero  [ Sat Jan 28, 2012 9:31 pm ]
Post subject:  Subtraction and division cages

Does the solution to a subtraction or division cage ever depend on the order of operations? For example, a 3-cell subtraction with a result of 1: the solution could be 4,2,1, as in 4 - 2 - 1 = 1. Can the answer ever be 4,4,1, as in 4 - (4 - 1) = 1?

I don't think I have ever seen a solution that depends on order of operations, but since I am never sure, subtraction and division cages with 3 or more cells always slow me down.

Author:  pnm  [ Sat Jan 28, 2012 9:39 pm ]
Post subject:  Re: Subtraction and division cages

rhpzero wrote:
Does the solution to a subtraction or division cage ever depend on the order of operations? For example, a 3-cell subtraction with a result of 1: the solution could be 4,2,1, as in 4 - 2 - 1 = 1. Can the answer ever be 4,4,1, as in 4 - (4 - 1) = 1?

I don't think I have ever seen a solution that depends on order of operations, but since I am never sure, subtraction and division cages with 3 or more cells always slow me down.

No, the order doesn't matter, or in other words, "there is an ordering for which the result is 1" (for your example).

Parentheses are never used.

Patrick

Author:  mparisi  [ Sun Jan 29, 2012 3:18 am ]
Post subject:  Re: Subtraction and division cages

Here's a trick to use with subtraction cages: the largest number in the cage is equal to the sum of the rest of the cells plus the number in the formula. Note, however, this doesn't work for puzzles with negative numbers, since the result of a subtraction doesn't necessarily decrease the running total.

Similarly for division cages the largest number in the cage is equal to the product of the rest of the cells and the number in the formula.

Author:  sneaklyfox  [ Sun Jan 29, 2012 4:02 am ]
Post subject:  Re: Subtraction and division cages

For the puzzles with negative numbers, for a 2-cell subtraction cage, I just think of the difference between the two numbers instead of actually calculating mathematically something like -2-(-1)=-1.

Author:  starling  [ Sun Jan 29, 2012 5:57 pm ]
Post subject:  Re: Subtraction and division cages

sneaklyfox wrote:
For the puzzles with negative numbers, for a 2-cell subtraction cage, I just think of the difference between the two numbers instead of actually calculating mathematically something like -2-(-1)=-1.


I still haven't figured out what to do with a 3-cell, though, to be honest.

Author:  sneaklyfox  [ Sun Jan 29, 2012 10:18 pm ]
Post subject:  Re: Subtraction and division cages

starling wrote:
sneaklyfox wrote:
For the puzzles with negative numbers, for a 2-cell subtraction cage, I just think of the difference between the two numbers instead of actually calculating mathematically something like -2-(-1)=-1.


I still haven't figured out what to do with a 3-cell, though, to be honest.


Leave it to the end. [razz]

Author:  starling  [ Mon Jan 30, 2012 12:14 am ]
Post subject:  Re: Subtraction and division cages

sneaklyfox wrote:
starling wrote:
sneaklyfox wrote:
For the puzzles with negative numbers, for a 2-cell subtraction cage, I just think of the difference between the two numbers instead of actually calculating mathematically something like -2-(-1)=-1.


I still haven't figured out what to do with a 3-cell, though, to be honest.


Leave it to the end. [razz]

Yep. That's literally what I've been having to do.

I had figured the highest number would still come first, but I've found some cases where it definitely doesn't.

Author:  threevalued  [ Fri Nov 22, 2013 10:25 am ]
Post subject:  Re: Subtraction and division cages

The assertion that "parentheses are never used" is not strictly correct. An expression such as '3-2-1' is straightforwardly ambiguous between (3-(2-1)) and ((3-2)-1); this is serious because subtraction is not associative (that is, the two unambiguous terms have different values, 2 and 0 respectively). If there is an implicit assumption of association to the left (in the example, that only the second disambiguation is correct), this should be made explicit in the rules. An example from yesterday's difficult 6x6: a three-cell was to have operation - and value 3. If we write this as x-y-z, where x, y and z can have values 1...6, AND assume association to the left, only 6-1-2 and 6-2-1 have value 3, so the cage contains 1, 2 and 6. But if we don't, then 6-(5-2), 4-(6-5) and many other terms have value 3, so we lose control of the cage. My point is simply that reading 6-5-2 as (6-5)-2 and hence having value -1 is just as much a use of parentheses as reading it as 6-(5-2).

Author:  calcpnm  [ Thu Dec 19, 2013 10:31 am ]
Post subject:  Re: Subtraction and division cages

threevalued wrote:
The assertion that "parentheses are never used" is not strictly correct. An expression such as '3-2-1' is straightforwardly ambiguous between (3-(2-1)) and ((3-2)-1); this is serious because subtraction is not associative (that is, the two unambiguous terms have different values, 2 and 0 respectively). If there is an implicit assumption of association to the left (in the example, that only the second disambiguation is correct), this should be made explicit in the rules

Apologies for the late reply:

Yes, there is such an assumption (evaluation from left to right).

Using parentheses can potentially change the ordering.

Note that in "kenken" puzzles cages
with subtraction or division are often restricted to 2 cells to avoid confusion.

Patrick

Author:  jpassaro  [ Mon Jan 20, 2014 5:24 am ]
Post subject:  Re: Subtraction and division cages

Apologies if this repeats what's been written elsewhere -- it's not clear for me from this post.

The criteria for a subtraction 3-cage, say "4-", is that there is some ordering (a,b,c) of the numbers where a-b-c = 4.

Must it be positive 4, or can it potentially be -4 as well?

For example, 0-1-3 = -4; could we use (0,1,3) as a solution for this cage?

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