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 April 1st, 2017, 12:27 PM #1 Newbie   Joined: Apr 2017 From: New York Posts: 1 Thanks: 0 Math game King Arthur wishes to construct a new shield by attaching square pieces with sizes 10×10 and 8×8 so that the corner of the smaller square is placed on the diagonal of the larger square and 3 units away from its center. King Arthur wants the smaller square to be sloped so that the area of overlap between the two squares is as small as possible since this will give a shield of the largest possible area. What is the area of King Arthur’s new shield? (Give a full explanation for your answer.) Last edited by skipjack; April 1st, 2017 at 07:25 PM.
 April 3rd, 2017, 01:12 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315 I'm showing the overlap area is independent of the rotation angle for angles $-\pi/4 \leq \theta \leq \pi/4$ Surprised by it myself but that's what it's looking like.
 April 3rd, 2017, 02:45 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,653 Thanks: 2086 Squares.jpg There are two diagrams that would be relevant. One is given above. In the other, PQ intersects the square ABCD.
April 3rd, 2017, 03:24 AM   #4
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Quote:
 Originally Posted by skipjack Attachment 8652 There are two diagrams that would be relevant. One is given above. In the other, PQ intersects the square ABCD.
this diagram doesn't show the corner of the smaller square along the diagonal of the larger.

One bit of ambiguity in the OP is that it doesn't specify whether the smaller corner is pinned to 3 in the nw direction or the se direction of the center of the larger.

I assumed the se direction.

April 3rd, 2017, 04:04 AM   #5
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Quote:
 Originally Posted by romsek this diagram doesn't show the corner of the smaller square along the diagonal of the larger.
Oh crap! So you're saying that CRAP should be a straight line?

April 3rd, 2017, 04:28 AM   #6
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Quote:
 Originally Posted by romsek This diagram doesn't show the corner of the smaller square along the diagonal of the larger.
That was intentional, as the problem description doesn't require it.

April 3rd, 2017, 01:09 PM   #7
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Quote:
 Originally Posted by mathpassion King Arthur wishes to construct a new shield by attaching square pieces with sizes 10×10 and 8×8 so that the corner of the smaller square is placed on the diagonal of the larger square and 3 units away from its center. King Arthur wants the smaller square to be sloped so that the area of overlap between the two squares is as small as possible since this will give a shield of the largest possible area. What is the area of King Arthur’s new shield? (Give a full explanation for your answer.)
I'm curious then what your interpretation of the bolded text is

there are admittedly 4 interpretations as to where this pivot point might be but they will all end up having equivalent answers as they are all the same problem but in a rotated coordinate system.

Do you have mathematica? I'd love a 2nd pair of eyes on this sheet.

 April 3rd, 2017, 07:56 PM #8 Global Moderator   Joined: Dec 2006 Posts: 20,653 Thanks: 2086 Squares2.PNG You're right. I was interrupted when I first read the question, having got as far as just making the larger square. A few days later, I came across the unfinished diagram and decided to finish it, but I misread the description.
April 3rd, 2017, 07:59 PM   #9
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The area of the quadrilateral outlined in black is

$\dfrac{59}{2} - 15 \sqrt{2}$

which is independent of $\theta$
Attached Images
 Clipboard01.jpg (74.7 KB, 15 views) Clipboard02.jpg (52.3 KB, 15 views)

 April 3rd, 2017, 08:10 PM #10 Global Moderator   Joined: Dec 2006 Posts: 20,653 Thanks: 2086 It's easy to see that a tilt angle of magnitude up to 45° doesn't affect the area of overlap. Hence the minimum overlap area is (5 - 3/√2)² (square units).

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