Dog's Mead - Futility Closet

dog's mead puzzle

This puzzle was devised by W.T. Williams. The goal is to discover the age of Father Dunk’s mother-in-law (2 down), but the clues contain so much cross-reference and the grid so many interlocking solutions that practically the whole puzzle must be completed to find it.

The year is 1939. There are 20 shillings in a pound; 4840 square yards in an acre; a quarter of an acre in a rood; and 1760 yards in a mile.

ACROSS

1. Area in square yards of Dog’s Mead
5. Age of Martha, Father Dunk’s aunt
6. Difference in yards between length and breadth of Dog’s Mead
7. Number of roods in Dog’s Mead times 8 down
8. The year the Dunks acquired Dog’s Mead
10. Father Dunk’s age
11. Year of Mary’s birth
14. Perimeter in yards of Dog’s Mead
15. Cube of Father Dunk’s walking speed in miles per hour
16. 15 across minus 9 down

DOWN

1. Value in shillings per rood of Dog’s Mead
2. Square of the age of father Dunk’s mother-in-law
3. Age of Mary, father Dunk’s other daughter
4. Value in pounds of Dog’s Mead
6. Age of Ted, father Dunk’s son, who will be twice the age of his sister Mary in 1945
7. Square of the breadth of Dog’s Mead
8. Time in minutes it takes Father Dunk to walk 4/3 times round Dog’s Mead
9. The number which, when multiplied by 10 across, gives 10 down
10. See 9 down
12. Addition of the digits of 10 down plus 1
13. Number of years Dog’s Mead has been in the Dunk family

Hint: Start with 15 across, and keep your wits about you — most of the clues require some sort of insight or intelligent narrowing of the possible solutions; there’s very little mechanical plugging of numbers. With enough careful, dogged reasoning, it’s possible to complete the entire grid, but it’s stupendously hard.

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Chris Adams sent me the guided solution below. Throughout, L is the length, B the breadth, and all units are in yards/square yards unless otherwise noted. Both 8 and 11 Across are years before 1939; since both are four digits, both must begin with a 1. The only two-digit cubes are 27 and 64, so 15 Across must be one of these numbers, and hence Father Dunk’s walking speed in miles per hour is either three or four. Let S denote his speed in miles per hour; then 88S/3 is his speed in yards per minute. And thus the time it takes him to walk the perimeter is 2(L+B)/(88S/3) = 3(L+B)/(44S), and so 4/3 this time (8 down) is (L+B)/(11S), and this is between 10 and 19 inclusive. Now, the perimeter is less than 1000 (since 14 Across is a three-digit number), so L+B, which is one half the perimeter, is less than 500. So L+B is equal to 11 times S times some number between 10 and 19, and this is lower than 500. Regardless of the value of S, 16 to 19 give us a value greater than 500, so the time is between 10 and 15 minutes. Look at 9 Down, 10 Across, and 10 Down. Since 9 Down ends in a 1, and 9 Down times 10 Across equals 10 Down, this means that 10 Across and 10 Down end in the same number. This also means that 8 Down ends with the same number that 15 Across begins with. And 15 Across begins with either 2 or 6, while 8 Down ends with 0, 1, 2, 3, 4, or 5. Hence that number must be a 2, so it takes 12 minutes, and the speed is 3 mph, which yields 27 as a cube. We know the speed and the time it takes to go 4/3 of the way around the perimeter, so we can solve for the perimeter and get that the perimeter is 792 yards. Alternatively, one can get that from L+B = 11×3×12, and P=2(L+B). Now, 12 Down is the sum of the digits in 10 Down, plus 1. This must be 19, as any other two-digit number ending in 9 is greater than any possible sum of the digits. Hence 10 Down must be 792, 882, or 972, making 10 Across 72, 82, or 92, respectively. Only in the first case does 10 Across divide 10 Down, since 792 equals 72×11. Hence 10 Down is 792, 10 Across is 72, and 9 Down is 11. This also makes 16 Across 16. 10 Down is a square and ends in 76, so B ends in either a 4 or a 6. Also, since L-B is less than 100, and L+B equals 396, B must be between 150 and 190, so there are eight possibilities for B. Of these, only 174 and 176 end in 76. However, 174 ends in 276, meaning Mary was born in the 1200s, which is plainly false (her father is only 72). Hence 76, which ends in 976, is the correct answer. The breadth B is 176, so the length L is 220, and this makes the area (1 Across) 38720. It also makes the number of roods 32, so 7 Across is 384. Also, 6 Across is 44, and so 6 Down is 48, which is Ted’s age. In six years, he’ll be 54, which is twice Mary’s age then. So she’ll be 27 then, which makes her 21 (3 Down) now. It also means she was born in 1918 (11 Across). Putting 8 Across and 13 Down together, we can fill in the blanks and show that the family acquired the property in 1110, so they’ve owned it for 829 years. 2 Down is a square number ending in 1, so the original number ends in 1 or 9. It is also in the 7000s, and since 80 squared is 6400 and 90 squared is 8100, this makes Father Dunk’s mother-in-law 81 or 89. 81 doesn’t work, but 89 squared is 7921, and so the mother-in-law is 89. 1 Down and 4 Down are still open, but we know that 1 Down is 5/8 (or 20/32, since there are 32 roods) of 4 Down. And 1 Down is between 300 and 400, so 4 Down must be between 480 and 640. Since 4 Down ends in 44, the only choice is 544, which makes 1 Down 340. This puzzle inspired an even harder one by CUNY math professor William Sit.

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dog's mead solution

Chris Adams sent me the guided solution below. Throughout, L is the length, B the breadth, and all units are in yards/square yards unless otherwise noted.

Both 8 and 11 Across are years before 1939; since both are four digits, both must begin with a 1.
The only two-digit cubes are 27 and 64, so 15 Across must be one of these numbers, and hence Father Dunk’s walking speed in miles per hour is either three or four. Let S denote his speed in miles per hour; then 88S/3 is his speed in yards per minute. And thus the time it takes him to walk the perimeter is 2(L+B)/(88S/3) = 3(L+B)/(44S), and so 4/3 this time (8 down) is (L+B)/(11S), and this is between 10 and 19 inclusive.
Now, the perimeter is less than 1000 (since 14 Across is a three-digit number), so L+B, which is one half the perimeter, is less than 500. So L+B is equal to 11 times S times some number between 10 and 19, and this is lower than 500. Regardless of the value of S, 16 to 19 give us a value greater than 500, so the time is between 10 and 15 minutes.
Look at 9 Down, 10 Across, and 10 Down. Since 9 Down ends in a 1, and 9 Down times 10 Across equals 10 Down, this means that 10 Across and 10 Down end in the same number. This also means that 8 Down ends with the same number that 15 Across begins with. And 15 Across begins with either 2 or 6, while 8 Down ends with 0, 1, 2, 3, 4, or 5. Hence that number must be a 2, so it takes 12 minutes, and the speed is 3 mph, which yields 27 as a cube.
We know the speed and the time it takes to go 4/3 of the way around the perimeter, so we can solve for the perimeter and get that the perimeter is 792 yards. Alternatively, one can get that from L+B = 11×3×12, and P=2(L+B).
Now, 12 Down is the sum of the digits in 10 Down, plus 1. This must be 19, as any other two-digit number ending in 9 is greater than any possible sum of the digits. Hence 10 Down must be 792, 882, or 972, making 10 Across 72, 82, or 92, respectively. Only in the first case does 10 Across divide 10 Down, since 792 equals 72×11. Hence 10 Down is 792, 10 Across is 72, and 9 Down is 11. This also makes 16 Across 16.
10 Down is a square and ends in 76, so B ends in either a 4 or a 6. Also, since L-B is less than 100, and L+B equals 396, B must be between 150 and 190, so there are eight possibilities for B. Of these, only 174 and 176 end in 76. However, 174 ends in 276, meaning Mary was born in the 1200s, which is plainly false (her father is only 72). Hence 76, which ends in 976, is the correct answer.
The breadth B is 176, so the length L is 220, and this makes the area (1 Across) 38720. It also makes the number of roods 32, so 7 Across is 384. Also, 6 Across is 44, and so 6 Down is 48, which is Ted’s age. In six years, he’ll be 54, which is twice Mary’s age then. So she’ll be 27 then, which makes her 21 (3 Down) now. It also means she was born in 1918 (11 Across).
Putting 8 Across and 13 Down together, we can fill in the blanks and show that the family acquired the property in 1110, so they’ve owned it for 829 years.
2 Down is a square number ending in 1, so the original number ends in 1 or 9. It is also in the 7000s, and since 80 squared is 6400 and 90 squared is 8100, this makes Father Dunk’s mother-in-law 81 or 89. 81 doesn’t work, but 89 squared is 7921, and so the mother-in-law is 89.
1 Down and 4 Down are still open, but we know that 1 Down is 5/8 (or 20/32, since there are 32 roods) of 4 Down. And 1 Down is between 300 and 400, so 4 Down must be between 480 and 640. Since 4 Down ends in 44, the only choice is 544, which makes 1 Down 340.

This puzzle inspired an even harder one by CUNY math professor William Sit.