4. Among 49 visually identical coins - 25 are genuine and 24 are counterfeit. To identify the counterfeit coins, there is a tester. You can place any number of coins in it, and if more than half of these coins are counterfeit, the tester will signal. How can you find two counterfeit coins in five tests? (K. Knop)
Solution. Let's call "working" those coins among which we continue to search for a pair of fake ones. We will denote the situation "N coins are working, among which there are at least M fake ones" as $N: M$. If the current test gives a signal (let's denote this as "+"), then after this, we consider working only those working coins that participated in the test, and if there was no signal ("-"), we continue to consider working only those working coins that did not participate in this test. Initially, we have the situation 49:24. Let's provide a possible algorithm for solving the problem.

Test 1: test 27 coins. +: situation 27:14, -: situation 22:11. Test 2 for situation 27:14: test 16 coins. +: situation $16: 9$, -: situation 11:6. Test 2 for situation 22:11: test 16 coins. In both cases - situation $11: 6$.

Test 3 for situation 16:9: test 8 coins. In both cases — situation 8:5. Test 4 for situation 8:5: test 4 coins. In both cases - situation 4:3. Test 5 for situation 4:3: test two coins. In both cases - situation 2:2, which means we have found a pair of fake ones.

Test 3 for situation 11:6: test 8 coins. +: situation 8:5, tests 4 and 5 for which are indicated above; -: situation $3: 2$. Tests 4 and 5 for situation $3: 2$ - test one coin twice. Either both will turn out to be fake ("+"), or one will be real ("-"), then the other fake coin will be the third of the remaining ones.