minimal cost to get a square from four sides

Question

Suppose I have four sides, and I want to make a square out of them. Each of the sides can be decreased or increased, and a price needs to be payed for such an action: increasing or decreasing a side by one unit has a cost of one unit. What is the minimum cost required to get a square out of these sides?

I saw a solution where they sorted the sides in increasing order, $a \leq b \leq c \leq d$, and then to get answer they did $(c - b) + (c - a) + (d - c)$. Why does this work?

Step 1 decipher the real problem from the insane wording of the original problem. — marshal craft, Apr 13 '17 at 10:22
Given $4$ sides having lengths $a,b,c$ and $d$ and cost per unit change $x$. Then $a+a_0=b+b_0=c+c_0=d+d_0$, total cost $$X=x(a_0+b_0+c_0+d_0)$$. The question is to minimize $X$. — marshal craft, Apr 13 '17 at 16:45

score 0 · Answer 1 · answered Apr 13 '17 at 10:21

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What do you mean by cost? If the square has side length $a$, decreasing a side's length by 1 will yield an area of $(a)(a-1)$. Please be more specific and place a link to the video here, so we can better comprehend the phrasing and intent of your question.

answered Apr 13 '17 at 10:21

Toby Mak

16,827

It might not be a square to start, the idea is to play with the sides and obtain a square by either increasing or decreasing the sides. Suppose a = 5, b = 5, c = 5, d = 6 i can just decrease d by one and obtain a square with all sides of length 5 and this has minimal cost of one unit. – Chris Yates Apr 13 '17 at 10:26
What do you mean by cost in your question? – Toby Mak Apr 13 '17 at 10:27
Please edit your question to clear up the confusion about cost, what shape you are referring to, and the video link. – Toby Mak Apr 13 '17 at 10:28
@Toby Mark the question has been corrected. – marshal craft Apr 14 '17 at 14:59
Is the quadrilateral always a rectangle? – Toby Mak Apr 15 '17 at 00:21

score 0 · Answer 2 · answered Apr 14 '17 at 11:00

I think the minimum cost is achieved by selecting a length for the sides so that it is greater than the second shortest side and less than the second longest side.

Four sides can be ordered like so $ a \le b \le c \le d$. Then the cost is minimized if length of final square's side $b \le L \le c$.

minimal cost to get a square from four sides

2 Answers2