Sensibilly -T ™ scales were out of stock, so Gramazon filled the order with a lousy Lopside® from Irreturnable Irrationals International, Inc.73205080... How irredeemably irritating. Sigh, might as well unpack it anyway and just start weighing things.
Parts
• One factory-calibrated lopsided beam balance
• One unit weight,
weight =  1   unit
• One disposable weighing pan,
weight =  p   units
( p is an
irrational amount)
• Another disposable weighing pan,
weight =  p   units
• Another disposable weighing pan,
weight =  p   units
⋮
(infinitely many identical pans)
Specifications
•
The scale balances when nothing hangs from either beam’s hook.
•
The scale also balances with the unit weight on the left side
and one pan on the right.
•
Any given weight that’s a positive integer W units heavy
may be balanced by hanging one or more pans,
placing W in one of those pans,
and optionally hanging the unit weight as well.
•
Each pan is good for a single balancing only
and is then wastefully discarded.
•
The unit weight may be used again and again.
Action plan
Day 1.
Balance W =  1  unit   (then discard all used pans).
Day 2.
Balance W =  2  units (and again discard the pans)
Day 3.
Balance W =  3  units (and discard pans)
⋮
Day 30.
Balance W =  30  units (discard pans one last time)
Among positive irrational numbers, what value for p allows the fewest pans total to be used up during these 30 action-packed days?
Notes
All items may hang on either side, but W always needs a pan on the same side to hang in.
Although the balance’s left beam is depicted above as being longer than its right beam, that disparity would be reversed if p were between 0 and 1, which is allowed.