In the Edo period, porridge eaten on the 15th day, namely "望の日" (mochi-no-hi: mochi written as 望 means a full moon, and mochi-no-hi means the day of the full moon, i.e. every 15th day of the lunar months) was somehow differently interpreted as porridge of "餅(の日)" (mochi(-no-hi): mochi written as 餅 means rice cake), and people started a new custom of eating azuki-gayu with rice cakes in it.
江戸時代には15日すなわち「望（もち）の日」の粥という語が転じて「餅（の日）」の粥と解せられ、小豆粥に餅に入れて食べる風習も行われるようになった。