story - The Knights and the Hounds

(With thanks and congratulations to Lee Leon)

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We rejoin our heroes several days after the events of The Knights' Gift, at the head of a pack of enormous staghounds, 1000 strong. Such a pack had never before been seen, and the barking and howling carried for many leagues.

It was a mighty pack, full of hounds of every color and pattern – 540 were solid, 2 for every hound that was mottled. 330 were brown but not spotted, and 120 were spotted but not brown.

Among the white dogs, twice as many were solid as were mottled, and half again were mottled as were spotted. And among the 380 grey animals, half as many were spotted as were mottled.

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Said one knight, “Did you really have to ask for 1000 enormous staghounds? We shall surely go broke feeding them, let alone finding a suitable place for them to live!

“Fear not!”, the other replied. “For I am in possession of a large quantity of land. We can divide it into 100 plots – 10 on each side – and there will be ample space for 10 staghounds per plot, even enormous as they may be. We shall divide the 100 plots into 9 rectangular pens, so that the animals may be sorted according to color and pattern.”

“Well, enough space is one thing, but how shall we arrange it? After all, not all the hounds get along – we must avoid letting any of the spotted dogs share fences with any of the mottled dogs. Well… except for the white spotted dogs, which seem to get along with the mottled dogs just fine.”

“Yes, that’s an excellent plan! And we must keep their special needs in mind, as well. The solid brown hounds should have at least 2 fence sections along the Southern edge, so that they can feel the sun on their fur, and the solid white hounds need at least 2 fence sections along the Eastern edge, so that they may feel the wind at their backs. And lastly, the grey solid and grey mottled animals are very friendly with each other and should share at least one fence section.”

“Very well, but with so many constraints, are you sure there’s a way to arrange them all?”

The knights stopped for a few moments, considering, then one opened a piece of parchment and began to sketch…

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Hint #1

Solid brown has exactly 4 fence sections along the southern border. Solid white has exactly 2 fence sections along the eastern border. Grey solid/grey mottled share exactly 6 fence sections.

Answer

With the help of Hint 1, here's the full answer.

First, the number of each kind of hound.

Hound table | brown | white | grey | total solid | 280 | 120 | 140 | 540 mottled | 50 | 60 | 160 | 270 spotted | 70 | 40 | 80 | 190 total | 400 | 220 | 380 | 1000

We start by finding pattern totals. solid = 540, mottled = solid/2 and spotted is what's left. 120 are spotted not brown so there are 190-120 = 70 that are spotted and brown. There are 70+330 = 400 total brown, and since grey = 380, white = 220. Then (white solid) = 2*(white mottled), (white mottled) = 1.5*(white spotted), so white = 220 = 2*(white mottled) + (white mottled) + 2/3*(white mottled) = 11/3*(white mottled) therefore, white mottled = 60. Following white spotted = 40 and white solid = 120. grey spotted = 190-70-40 = 80, then doubled for grey mottled = 160. The rest can be figured out be subtracting the other two from the total.

Possible pen dimensions

Now that we have the totals we can find all the possible pen dimensions. The longest side must be 10 or lower to fit on the land. Here are the possibilities.

solid brown = 4x7
solid white = 2x6, 3x4
solid grey = 2x7
mottled brown = 1x5
mottled white = 1x6, 2x3
mottled grey = 2x8, 4x4

spotted brown = 1x7
spotted white = 1x4, 2x2
spotted grey = 1x8, 2x4

Taking Hint 1 into account we can further reduce the following:

solid white = 2x6
mottled grey = 2x8

Pen arrangement

I found the arrangement with simple trial and error and a few good guesses.

solid brown = A
solid white = B
solid grey = C

mottled brown = D
mottled white = E
mottled grey = F
spotted brown = G
spotted white = H
spotted grey = I
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