Henry Ernest Dudeney/Modern Puzzles/80 - The Flocks of Sheep/Solution
Modern Puzzles by Henry Ernest Dudeney: $80$
- The Flocks of Sheep
- Four brothers were comparing the number of sheep that they owned.
- It was found that Claude had ten more sheep than Dan.
- If Claude gave a quarter of his sheep to Ben,
- then Claude and Adam would together have the same number as Ben and Dan together.
- If, then, Adam gave one-third to Ben,
- and Ben gave a quarter of what he then held to Claude,
- who then passed on a fifth of his holding to Dan,
- and then Ben divided one-quarter of the number he then possessed equally among Adam, Claude and Dan,
- they would all have an equal number of sheep.
- How many sheep did each possess?
Solution
Adam had $60$ sheep.
Ben had $50$ sheep.
Claude had $40$ sheep.
Dan had $30$ sheep.
Proof
Let $A$, $B$, $C$ and $D$ be the number of sheep owned by Adam, Ben, Claude and Dan respectively.
Let $B_1$ and $C_1$ be the number of sheep owned by Ben and Claude respectively after Claude gives Ben a quarter of his.
Let $B_2$ and $A_2$ be the number of sheep owned by Ben and Adam respectively after Adam gives Ben a third of his.
Let $B_3$ and $C_3$ be the number of sheep owned by Ben and Claude respectively after Ben gives Claude a quarter of his.
Let $C_4$ and $D_4$ be the number of sheep owned by Claude and Dan respectively after Claude gives Dan a fifth of his.
Let $A_5$, $B_5$, $C_5$ and $D_5$ be the number of sheep owned by Adam, Ben, Claude and Dan respectively at the end of the series of exchanges.
We have:
\(\ds C\) | \(=\) | \(\ds D + 10\) | It was found that Claude had ten more sheep than Dan. | |||||||||||
\(\ds B_1\) | \(=\) | \(\ds B + \dfrac C 4\) | If Claude gave a quarter of his sheep to Ben, | |||||||||||
\(\ds C_1\) | \(=\) | \(\ds \dfrac {3 C} 4\) | ||||||||||||
\(\ds C_1 + A\) | \(=\) | \(\ds B_1 + D\) | then Claude and Adam would together have the same number as Ben and Dan together. | |||||||||||
\(\ds B_2\) | \(=\) | \(\ds B_1 + \dfrac A 3\) | If, then, Adam gave one-third to Ben, | |||||||||||
\(\ds A_2\) | \(=\) | \(\ds \dfrac {2 A} 3\) | ||||||||||||
\(\ds B_3\) | \(=\) | \(\ds \dfrac {3 B_2} 4\) | and Ben gave a quarter of what he then held to Claude, | |||||||||||
\(\ds C_3\) | \(=\) | \(\ds C_1 + \dfrac {B_2} 4\) | ||||||||||||
\(\ds C_4\) | \(=\) | \(\ds \dfrac {4 C_3} 5\) | who then passed on a fifth of his holding to Dan, | |||||||||||
\(\ds D_4\) | \(=\) | \(\ds D + \dfrac {C_3} 5\) | ||||||||||||
\(\ds A_5\) | \(=\) | \(\ds A_2 + \dfrac {B_3} {12}\) | ||||||||||||
\(\ds B_5\) | \(=\) | \(\ds \dfrac {3 B_3} 4\) | and then Ben divided one-quarter of the number he then possessed equally among Adam, Claude and Dan, | |||||||||||
\(\ds C_5\) | \(=\) | \(\ds C_4 + \dfrac {B_3} {12}\) | ||||||||||||
\(\ds D_5\) | \(=\) | \(\ds D_4 + \dfrac {B_3} {12}\) | ||||||||||||
\(\ds A_5\) | \(=\) | \(\ds B_5 = C_5 = D_5\) | they would all have an equal number of sheep. | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {3 B_3} 4\) | \(=\) | \(\ds A_2 + \dfrac {B_3} {12}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds C_4 + \dfrac {B_3} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds D_4 + \dfrac {B_3} {12}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A_2\) | \(=\) | \(\ds \dfrac {3 B_3} 4 - \dfrac {B_3} {12}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {9 B_3} {12} - \dfrac {B_3} {12} = \dfrac {8 B_3} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 B_3} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds C_4 = D_4\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C_3\) | \(=\) | \(\ds \dfrac {5 C_4} 4\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 5 4 \dfrac {2 B_3} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {10 B_3} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 B_3} 6\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds B_2\) | \(=\) | \(\ds \dfrac {4 B_3} 3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds C_1\) | \(=\) | \(\ds C_3 - \dfrac {B_2} 4\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 B_3} 6 - \dfrac 1 4 \dfrac {4 B_3} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 B_3} 6 - \dfrac {2 B_3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 B_3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {B_3} 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds D\) | \(=\) | \(\ds D_4 - \dfrac {C_3} 5\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 B_3} 3 - \dfrac 1 5 \dfrac {5 B_3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 B_3} 6 - \dfrac {B_3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 B_3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {B_3} 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \dfrac {3 A_2} 2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 3 2 \dfrac {2 B_3} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds B_3\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds \dfrac {4 C_1} 3\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 4 3 \dfrac {B_3} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 B_3} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 D} 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds D + 10\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds D\) | \(=\) | \(\ds \dfrac {4 D} 3 - 10\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac D 3\) | \(=\) | \(\ds 10\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds D\) | \(=\) | \(\ds 30\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds 40\) |
and so on.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $80$. -- The Flocks of Sheep
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $162$. The Flocks of Sheep