There are theories that hishi mochi was originally the hishi-hanabira mochi (lozenged-shaped flower petal mochi) eaten in the imperial court at New Year's, and that the original triangle shape was changed to a lozenge shape to pray for fertility and family prosperity because the seeds of the water caltrop (which is written in Japanese with the same character as lozenge) are highly fertile and for longevity based on the story of a hermit who lived for a thousand years by eating the seeds.
宮中で正月に食べられる菱葩餅が起源である説や、元は三角形であったが菱の繁殖力の高さから子孫繁栄と菱の実を食べて千年長生きをした仙人にちなんで長寿の願いを込め菱形にした説がある。