Following problem: "We have 100 metal balls, 51 of which are radioactive. Furthermore we have a balance which is designed in such a way that one sphere fits on each of its two plates. If a radioactive sphere is placed on both plates, a lamp lights up. a lamp lights up, otherwise the light remains off. With how many measurements can you be sure to find all 51 radioactive spheres among the 100 spheres? (A) 98 (B) 99 (C) 145 (D) 146 (E) 153"
The official solution is that you need a minimum of 145 Measurements in the following way: "Let's arrange the balls in pairs! In this case there is at least one es at least one pair consisting of two radioactive spheres. Now we put the two spheres of all 50 pairs on the scale. If the lamp lights up exactly once, then we have found exactly 2 radioactive spheres with 50 measurements. Furthermore, we know that in the remaining pairs always one sphere is radioactive. Now we take one of the found radioactive spheres and put one of the spheres of the remaining 49 pairs to it one after the other. If the lamp lights up, then we have found a radioactive sphere, if it does not light up, then the other sphere of the pair just considered is radioactive. Thus, with 50 + 49 = 99 measurements, we have found all 51 radioactive spheres. found. If the lamp lights up at least twice, then we have found 4 radioactive spheres. radioactive spheres have been found. Furthermore we know that among the remaining pairs, one pair certainly consists of non-radioactive spheres. So we have to all the balls of the remaining 48 pairs, except for the last ball. ball. Of this one we already know whether it is radioactive or not. or not. The number of measurements is: 50+(2*48-1)=50+95=145. Thus, we have to make at least 145 measurements to find for sure the 51 radioactive spheres."
We already have found a solution to the problem with only 124 tries, but we were wondering, if there are ways to go under 100 tries and by the way, how does this scale with a bigger amount of balls and are there formulas to determine the maximum number of tries for problem like this?
Thanks you for help.