$1.[$Adversarial design$]$

a$.$ You are given five coins; three are genuine, two are counterfeit. All are identical in appearance, but the counterfeit coins are either both heavier or both lighter than the other three. Using a pan balance, devise an algorithm to identify the counterfeit coins, including whether they are heavy or light. You may use the balance only three times.

Note: If there is a weight difference in the two pans, the balance will tip at the same rate regardless of the weight difference.

Hint: Balance is the key!

b$.$ Suppose there are only four coins with one counterfeit coin that is either heavy or light. Explain why you cannot find the counterfeit and its heavy$/$light status using a pan balance twice.