This is problem 23 on the grades 11 and 12, 2025, Math Kangaroo. A group of three square men from Mars and a group of three circular men from Jupiter are sitting around a table as shown. One of these six has the key to their flying saucer. All members of one group always tell the truth and all members of the other group always lie. All six were asked the question, Does a person sitting next to you have the key? Their answers are shown in the figure. Who has the key? The answers are A, Alien A has it, B, B has it, C, C has it, D, D has it, and E, E has it. I'm going to take your time, pause the video, try and solve it for yourself. Okay, so immediately the question sort of tells us that there are two cases. Either the squares are truth-tellers and the circles are liars or the other way around. Circles are truth and squares lie. So then let's like actually just work with both of these cases. So we have case one and we have case two. So let's call this one, you know, circles are truth and this one will be squares are truth. And then let's just like go about the video or go about each thing. So if circles are telling the truth, if the circle is telling the truth and like the answer is yes, then one of these two has the key. Similarly, if this circle is telling the truth, then one of these two has the key. Really the only place of overlap is at this yes here. So that sort of tells us that, yeah, the key should be right here. And let's see if there are like, if there's any any truth to be had or like any other contradictions in here. So if the key is right there, then all the squares need to be lying. However, if the square were lying, then this should have been a yes. Similarly, this should have been a yes. And like this yes is valid because that alien has the key. And thus the key is not like next to him. Okay, so we found a few contradictions there. Let's try the other case. So similarly, yeah, we have that this alien is telling, we have that the squares are telling the truth. And therefore if this alien answers yes, then one of these two has the key. And let's go, let's analyze another truth-telling alien. This one's telling the truth with no. So that means neither of these two have the key. And therefore like this alien right here should have the key. So now let's check on whether or not the circles are lying and whether or not the squares are also lying. So this square, this square is telling the truth. So it's like, it's valid. Because neither of these has the key. This circle is lying because his neighbor does have the key. So that checks out. This circle is lying because he has the key and one of his neighbors does not have a key. And we've already sort of like based our assumptions on these two telling the truth. And finally, neither of these has the key. So this bottom left alien is lying. So we've gotten six check marks. So that means this second configuration is valid. And therefore alien B has the key. So our answer is B.

aliens

truth-tellers

liars

key

puzzle