Sunzi Suanjing Middle Scroll 25

## **Problem**

答曰：公一十八顆。侯一十五顆。伯一十二顆。子九顆。男六顆。

術曰：先置人數，別加三顆於下，次六顆，次九顆，次一十二顆，上十五顆。副并之，得四十五。以減六十顆，餘，人數除之，人得三顆。各加不并者，上得一十八，為公分；次得一十五，為侯分；次得十二，為伯分；次得九，為子分；下得六，為男分。

Answer: Duke 18, Marquess get 15, Earl 12, Viscount 9, and Baron 6.

Method: Begin by setting the number of people with 3 mandarin oranges at the bottom rank, which is the amount to add for each successive rank: 6 in the next rank, 9 in the next rank, 12 in the next rank, and 15 in the next rank. Add them all to obtain 45. Subtract this from 60 and divide the remainder by the number of people to get 3 tangerines for each person. Add this amount to each number [in each rank] before they were summed to obtain 18 in the top rank, which is the Duke's share; 15 in the next rank, which is Marquess's share; 12 in the next rank, which is Count's share, 9 in the next rank which is Viscount's share; and 6 in the bottom rank, which is Baron's share.

## **Solution**

**Method 1: The method in the *Sunzi Suanjing***

Begin by setting up the following table

 Marquess
 Count
 Viscount
 Baron

Add all the numbers in the table.

3
6
9
12
# 15
45

{\displaystyle 3 + 6 + 9 + 12 + 15 = 45}

Subtract the total 45 from the total number of mandarin oranges 60 and then divide by the number of people 5.

60
−
45

5

=
3

{\displaystyle \frac{60-45}{5} = 3}

Add this result to each number in the original table to get the number of mandarin oranges for each nobleman.

 Marquess
 Count
 Viscount
 Baron

Comment: This method is essentially the same as Method 2 except they let the number of mandarins that the baron be

x
3

{\displaystyle x+3}

. Unknowns were not solved by introducing variables and writing down equations; hence, the solution method relies on building a table and performing the arithmetic instructions equivalent to solving an equation.

**Method 2: Algebraic method**

Let

x

{\displaystyle x}

 represent the number of mandarins that the baron has. Then we get the sequence

Baron -

x

{\displaystyle x}

Viscount -

x
3

{\displaystyle x+3}

Count -

x
6

{\displaystyle x+6 }

Marquess -

x
9

{\displaystyle x+9 }

Duke -

x
12

{\displaystyle x+12 }

Adding the number of mandarins each noble has yields:

x
(
x
3
)
(
x
6
)
(
x
9
)
(
x
12
# )
60

5
x
# 30
60

{\displaystyle \begin{align}
x+(x+3)+(x+6)+(x+9)+(x+12) = 60 \\[5 pt]
5x + 30 = 60
\end{align} }

Now solving for the unknown yields:

5

# x
6

{\displaystyle \begin{align}
5x = 30 \\[5 pt]
x = 6
\end{align} }

The baron has 6, the viscount has 9, the count has 12, the marquess has 15, and the duke has 18.

**Method 3: Arithmetic series**

This problem can also be solved using the formula for calculating arithmetic series.

S
n

=

n
2

[
2
a
(
n
−
1
)
d
]

{\displaystyle S\_n = \frac{n}{2} [2a + (n-1)d]}

For this problem, the sum is

S
n

=
60

{\displaystyle S\_n = 60 }

, the number of terms is

# n
5

{\displaystyle n=5}

, the first term is

a

{\displaystyle a}

, and the common difference is

# d
3

{\displaystyle d=3}

. Substitute the known values and solve for the first term.

# 60

5
2

[
2
a
(
5
−
1
)
3
]

{\displaystyle 60 = \frac{5}{2} [2a + (5-1)3]}

# 24
2
a
12

{\displaystyle 24 = 2a + 12 }

# a
6

{\displaystyle a=6}

The arithmetic sequence is thus

6
,
9
,
12
,
15
,
18

{\displaystyle 6, 9, 12, 15, 18 }

, which corresponds with the number of mandarin oranges each nobleman of ascending rank receives.