an ancient device for working with numbers, count, calculate, figure
筭 centers on working with numbers using a physical tool: counting, calculating, or figuring. The ancient device sense is the concrete origin, while the other meanings are natural extensions of numerical work.
筭 combines 竹 (bamboo) with 弄 (play with, handle), suggesting a bamboo tool used for handling numbers. The exact historical development is uncertain, but the component relationship is consistent with an ancient counting device.
Bamboo 竹 over handling 弄: picture an old counting board made of bamboo strips that you handle to count and calculate.
For サン, imagine the sun (サン) shining on your bamboo counting tool as you figure out the day's tally: sun -> サン.