# interesting problem

#### the undertaker

is it possible to arrange integers from 1 to 240 in a table with 15 rows and 16 columns so that the sum of the numbers in each of the columns would be the same?

#### Opalg

MHF Hall of Honor
is it possible to arrange integers from 1 to 240 in a table with 15 rows and 16 columns so that the sum of the numbers in each of the columns would be the same?
Yes. For example, row 1 could consist of the numbers 1 to 15, with row 2 containing 240 to 226 (in decreasing order). Then row 3 has 16 to 30, row 4 has 225 to 211, and so on. The bottom two rows contain 102 to 120 and 135 to 121. In each of those pairs of rows, the two numbers in each column have sum 241. So the sum of the numbers in each complete column is 8×241 = 1928.

#### Soroban

MHF Hall of Honor
Hello, the undertaker!

Is it possible to arrange integers from 1 to 240 in a table with 15 rows and 16 columns
so that the sum of the numbers in each of the columns would be the same? . . . . no

The sum of the numbers is: .$$\displaystyle \frac{240\cdot241}{2} \:=\:28,\!920$$

Then each column would have a sum of: .$$\displaystyle \frac{28,\!920}{16} \:=\:1807\tfrac{1}{2}$$

• Opalg

#### Opalg

MHF Hall of Honor
Hello, the undertaker!

The sum of the numbers is: .$$\displaystyle \frac{240\cdot241}{2} \:=\:28,\!920$$

Then each column would have a sum of: .$$\displaystyle \frac{28,\!920}{16} \:=\:1807\tfrac{1}{2}$$

Quite right — I had rows and columns the wrong way round!