The Book of Numbers begins the year after the Israelites left Egypt. Most of that year has been spent at Mt. Sinai:
The LORD spoke to Moses in the wilderness of Sinai, in the tent of meeting, on the first day of the second month, in the second year after they had come out of the land of Egypt...
In Numbers 14:34, The Lord says to the Israelites that they will spend forty years in the wilderness, one year for each day the unfaithful spies were spying out the land.
Numbers 14:34 ESV
According to the number of the days in which you spied out the land, forty days, a year for each day, you shall bear your iniquity forty years, and you shall know my displeasure.
In Numbers 20:28 Aaron dies on Mount Hor. The date of his death is given in Numbers 33:38 as "the 40th year, first day, 5th Month."
Numbers 33:38 ESV
And Aaron the priest went up Mount Hor at the command of the LORD and died there, in the fortieth year after the people of Israel had come out of the land of Egypt, on the first day of the fifth month.
Thus the question: did the Israelites spend one year at Mount Sinai, and then an additional 40 years in the wilderness (thus 41 years in the wilderness), or is the first year at Mount Sinai included in the 40-year count?
Depending upon the answer to the above question, do Numbers 20 to 32 take place in a time span of six months or a year and six months?