快猫短视频

#207 Total recall

Set by Zoe Griffiths

I recently had the incredible opportunity of talking to an intergalactic traveller about her encounter with four extraterrestrial beings.

鈥淭he aliens have purple skin, long, floppy legs and large orange eyes,鈥� she told me. 鈥淎nd they revealed to me the true nature of dark matter.鈥�

I leaned in close with excitement.

鈥淯nfortunately, I don鈥檛 remember anything about that. I do remember a great puzzle they told me, though. I asked them their ages, and they said that if you sum up the ages of only three of them, the possible totals are 24, 53, 54 and 61. From this, they told me that I could work out all four of their ages.鈥�

Well, it isn鈥檛 exactly the secrets of the universe, but it will have to do. Can you work out the ages of the four aliens?

Solution next week

#206 All square

Solution

Lagrangia (L) scored 576 in their first innings; the Pythagoreans (P) managed 225. In their second innings, Lagrangia scored 49 and the Pythagoreans reached 400.

We need four square numbers such that L1 + L2 = P1 + P2. We know that L1 > 300. We also know L2 < 50 and L1 + L2 is a square number. So, we need two square numbers over 300 that differ by less than 50. Adjacent squares are separated by larger and larger odd numbers, and the first two over 300 are 37 apart. This means that 37 < L2 < 50. The only odd square in this range is 49. If L2 = 49, then L1 = 576 and P1 + P2 = 625. We also know P1 < 376 and P2 > 249. The only square numbers that fit these criteria are 225 and 400.