## Task 2 - 040512

According to the calibration regulations, in the range from $1 \mathrm{~g}$ to $1 \mathrm{~kg}$, only weights of the following sizes are permitted:

$1 \mathrm{~g}, 2 \mathrm{~g}, 5 \mathrm{~g}, 10 \mathrm{~g}, 20 \mathrm{~g}, 50 \mathrm{~g}, 100 \mathrm{~g}, 200 \mathrm{~g}, 500 \mathrm{~g}, 1 \mathrm{~kg}$.

A balance should be able to determine all mass values between $1 \mathrm{~g}$ and $2 \mathrm{~kg}$ in increments of $1 \mathrm{~g}$. The weights should only be placed on one side of the balance.

How many weights of the above types are then required if their total number is to be as small as possible?
With the pieces of $1 \mathrm{~g}, 2 \mathrm{~g}, 2 \mathrm{~g}, 5 \mathrm{~g}$, all values from $1 \mathrm{~g}$ to $9 \mathrm{~g}$ can be reached, with $10 \mathrm{~g}, 20 \mathrm{~g}, 20 \mathrm{~g}, 50 \mathrm{~g}$ all possible tens from $10 \mathrm{~g}$ to $90 \mathrm{~g}$, and with $100 \mathrm{~g}, 200 \mathrm{~g}, 200 \mathrm{~g}, 500 \mathrm{~g}$ all hundreds from 100 $\mathrm{g}$ to $900 \mathrm{~g}$. With the $1 \mathrm{~kg}$ weight, all values from $1 \mathrm{~kg}$ to $2 \mathrm{~kg}$ can be determined, meaning the following weights are needed:

$1 \mathrm{~g}, 2 \mathrm{~g}, 2 \mathrm{~g}, 5 \mathrm{~g}, 10 \mathrm{~g}, 20 \mathrm{~g}, 20 \mathrm{~g}, 50 \mathrm{~g}, 100 \mathrm{~g}, 200 \mathrm{~g}, 200 \mathrm{~g}, 500 \mathrm{~g}$ and $1 \mathrm{~kg}$.