Mahaviracharya/Ganita Sara Samgraha/Chapter VI/289

Mahaviracharya: Ganita Sara Samgraha Chapter $\text {VI}$: Mixed Problems: Problem $289$

Arrows, if they are thin cylinders, circular in cross-section, can be packed in hexagonal bundles.

Let the arrows around the circumference be $18$ in number.

How many in all are the arrows to be found in the bundle in the quiver?

Solution

$37$

Proof

The arrows are packed as centered hexagonal numbers.

The first few centered hexagonal numbers are as follows:

Sequence of Centered Hexagonal Numbers

The sequence of centered hexagonal numbers, for $n \in \Z_{> 0}$, begins:

$1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, \ldots$

The answer is immediately apparent.

$\blacksquare$

Sources

• c. 850: Mahaviracharya: Ganita Sara Samgraha: Chapter $\text {VI}$: Mixed Problems: $289$
• 1912: Rao Bahadur M. Rangacharya: The Ganita-Sara-Sangraha of Mahaviracharya: Chapter $\text {VI}$: Mixed Problems: $289$
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $53$