In the 1974 MAA Annual High School Mathematics Examination (Problem #9), students solved this grid pattern riddle:

"The integers greater than one are arranged in five columns as follows:

       Col 1  Col 2  Col 3  Col 4  Col 5
Row 1:          2      3      4      5
Row 2:   9      8      7      6       
Row 3:         10     11     12     13
Row 4:  17     16     15     14       

Four consecutive integers appear in each row. In the odd-numbered rows, the integers appear in the last four columns and increase from left to right. In the even-numbered rows, the integers appear in the first four columns and increase from right to left.

In which column will the number 1,000 fall? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5."

Let's examine the periodicity of the grid:

• Row 1 contains $4$ numbers ($2, 3, 4, 5$).
• Row 2 contains $4$ numbers ($6, 7, 8, 9$).

Together, every pair of rows contains $4 + 4 = 8$ numbers. Therefore, the pattern repeats every 8 numbers!

Let's see where numbers land based on their remainder modulo 8:

• $2 \pmod 8 = 2 \implies \text{Column 2}$
• $3 \pmod 8 = 3 \implies \text{Column 3}$
• $4 \pmod 8 = 4 \implies \text{Column 4}$
• $5 \pmod 8 = 5 \implies \text{Column 5}$
• $6 \pmod 8 = 6 \implies \text{Column 4}$
• $7 \pmod 8 = 7 \implies \text{Column 3}$
• $8 \pmod 8 = 0 \implies \text{Column 2}$
• $9 \pmod 8 = 1 \implies \text{Column 1}$

For the number $1,000$: $$1000 \div 8 = 125 \text{ remainder } 0$$ Since $1000 \equiv 0 \pmod 8$, it shares the same column as $8$, which is Column 2!

In this lesson, we write a simulation loop in Lua to populate the serpentine rows and locate the column of 1,000!