The two formulas doing the work
Paddock count is the classic rotation identity: paddocks = (rest days ÷ grazing days) + 1 — the +1 is the paddock the herd is standing in while the others rest. 30 days of rest at 3-day moves = 11 paddocks. Fractional answers always round up: 10.5 paddocks means 11, because rounding down shorts your rest period.
Paddock size comes from supply and demand: the herd needs (herd weight × intake %) pounds of dry matter per day, and each acre offers (standing DM × utilization). Days × demand ÷ per-acre supply = acres per paddock. Note that intake and utilization are separate numbers — intake is what the animals eat; utilization is the share of standing forage they can harvest before residual, rejection, and trampling stop them. Counting waste in both places double-counts it.
Rest is seasonal: cool-season pastures regrow fast in spring and slow in summer — many graziers run shorter rests in May and stretch them out past 40 days in August. Set the rest input to the season you're planning, not an annual average.