## 02 June 2023

### Math puzzle for you

A group of "space Marines" are returning from defending our planet against carnivorous alien invaders.  79 per cent of the Marines have lost an arm, 84 per cent have lost a leg, 76 per cent have lost an ear, and 71 per cent have lost an eye.  It clearly was one heck of a battle.

What percentage of these combatants, at the very least, must have lost all four body parts?

I have changed the circumstances and the numbers, because this math puzzle was first formulated in the 1800s by Carolus Lodovicus, a famous Oxfordian mathematician, and I didn't want you guys to Google the keywords.  If your answer is anything other than zero, you must explain why.  This is the final exam; the result counts for 100% of your grade for the semester.  Summer vacation begins tomorrow.

Credit for finding this goes to Anne Fadiman.

1. Answer encoded with base64 to really avoid spoilers. https://www.base64encode.com/
MTAlIEFzc3VtZSBhIGJvZHkgb2YgMTAwIHNvbGRpZXJzLCBmb3IgZWFzZSBvZiBjYWxjdWxhdGlvbiwgYW5kIHRoYXQgZWFjaCBzb2xkaWVyIGhhcyBsb3N0IGF0IGxlYXN0IDMgbGltYnMuIFRoaXMgd291bGQgbWVhbiB0aGF0IHRoZSBib2R5IGhhcyBzdWZmZXJlZCBhIGxvc3Mgb2YgNzkgYXJtcywgODQgbGVncywgNzYgZWFycywgYW5kIDcxIGV5ZXMsIGZvciBhIHRvdGFsIG9mIDMxMCBsaW1icyBmcm9tIDEwMCBzb2xkaWVycy4gMzAwIG9mIHRob3NlIHdvdWxkIGJlIGNvdmVyZWQgYnkgdHJpcGxlIGFtcHV0ZWVzLCBsZWF2aW5nIDEwIHF1YWRydXBsZSBhbXB1dGVlcy4=

2. If your answer is zero, that also needs to be justified!

3. Not being a math guy, logic tells me it can't be more than 71%.
xoxoxoBruce

4. 21% still have an arm, 16% still have a leg, 24% still have an ear, and 29% still have an eye. If these groups of Marines are all disjoint, then 90% still have a body part. In general, at most 90% still have a body part. Thus at least 10% have lost all parts.

5. I'm guessing 10: 71-(100-(76-(100-(84-(100-79))))). As proof, note that this equation works no matter how you rearrange the percentages. What is taking place is calculating the minimum overlap between any of the two percentages, then taking that result and calculating the minimum overlap between it and the next percentage, and the then the fourth.

6. Richard "RichiH" HartmannJune 2, 2023 at 4:57 PM

10%, as that is there smallest possible overlap. Take 100-x=y per limb, 100-y1-y2-y3-y4=z=10>

7. I did it Joe's way but slowly (whilst drawing the overlaps on paper):
21 have an arm and 16 have a leg so 100-21-16=63 have neither, so 37 have AorL
37 have AorL, 24 have an ear so 100-37-24=39 have neither, so 61 have AorLorE
61 have AorLorE, 29 have an eye so 100-61-29=10 have neither.

8. I hope they won that battle, because this army ain't never gonna fight again.

9. Just in case you haven't seen these other problems from "Carolus Lodovicus " https://math.hawaii.edu/~hile/math100/logice.htm

10. My answer is 0. I don’t have to explain it, but I can. Slight spoiler is that these numbers are not arbitrary. They were chosen to make the answer 0. An increase in any number would make the answer not 0.

11. I just want to know if the carnivorous alien invaders have been stopped...for good. Up to now, I've been for slashing the "defense" budget.

12. Cutting the defense budget is not a problem if it's officers golf courses and not carnivorous alien defense measures.
xoxoxoBruce

13. If everyone lost exactly 3 limbs, then these percentages must sum to at most 300%. But the sum is 310%, and so 10% must have lost a fourth limb. This is obviously attainable.

1. That's what I got!

If you think of the percentages as counts, then 310 parts have been lost: so at least 10 soldiers lost 4 parts.

14. This is a puzzle used in geocaching, with the numbers and circumstances altered.

The base 64 is also very common in geocaching puzzle caches.

Also from the creator if this math puzzle:

Ye golden hours of Life’s young spring,
Of innocence, of love and truth!
Bright, beyond all imagining,
Thou fairy-dream of youth!

I’d give all wealth that years have piled,
The slow result of Life’s decay,
To be once more a little child
For one bright summer-day.

15. > at the very least, must have lost all four body parts

So some marines have lost more than four body parts?

1. The commas designate that the phrase modifies "percentage of these combatants," not "body parts." But I think you knew that...

16. I was trying to determine if this is a math puzzle or (oh dread) a semantics puzzle. From your answer, I think it's intended to be mathematical.

17. I Googled it and got 10