In a group of 9 coins, one is fake and weighs slightly less. What is the fewest number of times you need to use a pan balance to find the counterfeit coins? (A pan balance is a scale used to measure weights, where weights are placed in two pans. The heaviest

pan will go down and the other pan will go up.)

9 Coins total and 8 are equal weight.

1st weight time=

(LEFT): 1,2,3 (Right): 4,5,6 7,8,9

2nd Weight time= 1,2 6,7 8,9

=2 times to use a pan.

1234 against 4567 if equal, 9 is fake

then 1234 against 5679
take th low side
xxafainst yy
take the low side
z vs other ...4 trials max

To find the counterfeit coin in a group of 9 coins, where one is fake and weighs slightly less, you need to use the pan balance scale.

- First, take three coins from one side of the scale and three coins from the other side. Place them on the scale, with three coins on each side.
(LEFT): 1,2,3 (Right): 4,5,6 7,8,9

- If the scale is balanced, then the counterfeit coin must be among coins 7, 8, or 9 (the ones that were not weighed).
In this case, take any two of these three coins and place them on the scale. If one side is lighter, that coin is the counterfeit one.
Otherwise, the remaining coin is the counterfeit.

- If the scale is not balanced, you'll need to perform another weighing. Take the lighter side of the previous measurement (containing the counterfeit coin) and divide it into two groups of two coins each.
Place one group on each side of the scale, leaving one coin aside.

(LEFT): 1,2 6,7 8,9

- If the scale is balanced this time, then the remaining coin is the counterfeit one.
- If the scale is not balanced, the lighter side contains the counterfeit coin. Take these two coins and place one on each side of the scale. The lighter side will indicate the counterfeit coin.

Therefore, the fewest number of times you need to use the pan balance to find the counterfeit coin among 9 coins is 2 times.