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Subtraction and division cages
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Posted on: Sat Jan 28, 2012 9:31 pm

Posts: 1
Joined: Thu Jan 12, 2012 8:49 pm
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.

Posted on: Sat Jan 28, 2012 9:39 pm

Posts: 2246
Joined: Thu May 12, 2011 11:58 pm
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

Posted on: Sun Jan 29, 2012 3:18 am

Posts: 93
Joined: Sat May 14, 2011 1:08 am
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.

Posted on: Sun Jan 29, 2012 4:02 am

Posts: 422
Joined: Fri May 13, 2011 2:43 am
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.

Posted on: Sun Jan 29, 2012 5:57 pm

Posts: 175
Joined: Fri May 13, 2011 2:11 am
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.

Posted on: Sun Jan 29, 2012 10:18 pm

Posts: 422
Joined: Fri May 13, 2011 2:43 am
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.

Posted on: Mon Jan 30, 2012 12:14 am

Posts: 175
Joined: Fri May 13, 2011 2:11 am
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.

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.

Posted on: Fri Nov 22, 2013 10:25 am

Posts: 1
Joined: Mon Nov 18, 2013 2:43 am
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).

Posted on: Thu Dec 19, 2013 10:31 am

Posts: 108
Joined: Wed May 11, 2011 6:44 pm
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

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

Posted on: Mon Jan 20, 2014 5:24 am

Posts: 1
Joined: Thu Dec 12, 2013 9:04 pm
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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