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# Circling cards

## Question

What is the radius of the smallest circle that can enclose all 52 non-overlapping cards of an ordinary deck of playing cards? And what is the configuration of the cards? Assume the smaller dimension of the cards is 5 and the larger is 7.

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Make 9 rows of cards with each row centered on the ones vertically adjacent to it:

Row 1: 3 cards abutted on their length 5 sides (row has length 21)
Row 2: 5 cards abutted on their length 5 sides (row has length 35)
Row 3: 6 cards abutted on their length 5 sides (row has length 42)
Row 4: 7 cards abutted on their length 5 sides (row has length 49)
Row 5: 10 cards abutted on their length 7 sides (row has length 50)
Row 6: 7 cards abutted on their length 5 sides (row has length 49)
Row 7: 6 cards abutted on their length 5 sides (row has length 42)
Row 8: 5 cards abutted on their length 5 sides (row has length 35)
Row 9: 3 cards abutted on their length 5 sides (row has length 21)

The radius of the smallest circle containing the cards is 25.9326.
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the best i can do

a 6x5 rectangle, with an additional 6 on the top and bottom, and 5 left and right.

gives a circle of radius 31.9

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Make 9 rows of cards with each row centered on the ones vertically adjacent to it:

Row 1: 3 cards abutted on their length 5 sides (row has length 21)

Row 2: 5 cards abutted on their length 5 sides (row has length 35)

Row 3: 6 cards abutted on their length 5 sides (row has length 42)

Row 4: 7 cards abutted on their length 5 sides (row has length 49)

Row 5: 10 cards abutted on their length 7 sides (row has length 50)

Row 6: 7 cards abutted on their length 5 sides (row has length 49)

Row 7: 6 cards abutted on their length 5 sides (row has length 42)

Row 8: 5 cards abutted on their length 5 sides (row has length 35)

Row 9: 3 cards abutted on their length 5 sides (row has length 21)

The radius of the smallest circle containing the cards is 25.9326.

I would like to see what you mean.

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Make 9 rows of cards with each row centered on the ones vertically adjacent to it:

Row 1: 3 cards abutted on their length 5 sides (row has length 21)

Row 2: 5 cards abutted on their length 5 sides (row has length 35)

Row 3: 6 cards abutted on their length 5 sides (row has length 42)

Row 4: 7 cards abutted on their length 5 sides (row has length 49)

Row 5: 10 cards abutted on their length 7 sides (row has length 50)

Row 6: 7 cards abutted on their length 5 sides (row has length 49)

Row 7: 6 cards abutted on their length 5 sides (row has length 42)

Row 8: 5 cards abutted on their length 5 sides (row has length 35)

Row 9: 3 cards abutted on their length 5 sides (row has length 21)

The radius of the smallest circle containing the cards is 25.9326.

I would like to see what you mean.

I wish I could make a picture of it, but I have never done anything like that. I've only used text.

Perhaps a text picture will help. Each 5x7 card is represented by a 5x7 matrix containing 35 instances of a single digit.

First and second rows:

```       111111122222223333333
111111122222223333333
111111122222223333333
111111122222223333333
111111122222223333333
44444445555555666666677777778888888
44444445555555666666677777778888888
44444445555555666666677777778888888
44444445555555666666677777778888888
44444445555555666666677777778888888```

I can't do the third row because the next card would have half of its 7-long side having blank space above it, and the

other half having half of card 4 above it.

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here's a picture of superprismatics solution.

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here's a picture of superprismatics solution.

Thats a nice picture, phil

That's a nice picture, phil. It's missing the two six-long rows (3rd & 7th). Would you add them?

I'm impressed that you could do such a good job on the two rows closest to the middle row, as

they are each 49 units long whereas the middle row is 50 -- making it difficult to center them.

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well done Phil. I was having a hard time visualizing what Superprismatic meant.

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