You are an army general trying to work out how many soldiers you need to send in to clear a city of zombies. To have any chance of taking the city back, past experience has told you that you need to have at least one soldier for every 10 zombies – anything less than that and you will not succeed. However, if you send in too many, you will leave your base without enough soldiers to protect it if zombies attack. This means you need to send in exactly the right number of soldiers – no more, no less. To get an estimate of the number of zombies in the city, you’ve sent out a helicopter to count the number of zombies in five randomly-selected city blocks. It has counted 523 zombies in the first one, 632 in the second, 781 in the third, 421 in the fourth and 307 in the fifth one. Given that for every one zombie seen on the streets you know there’s 4 hidden from sight in the buildings, and that there are 142 similar-sized blocks in the city, how many soldiers do you need to send in to guarantee that you’ll be able to take back the city without jeopardizing the safety of your base?

**A:** 2,738 soldiers

**B:** 15,627 soldiers

**C:** 37,829 soldiers

**D:** 65,342 soldiers

Scroll down to see the right answer…

**What answer did you get?**

**A:** With that many soldiers, there’d only be one for every 138 zombies and they’ll get massacred in seconds!

**B:** That’s closer to the right number, but it’s still not enough since you’d only have one solder for every 24 zombies. They’ll last longer but your troops will still lose in the end.

**C:** Spot on. You’ll have exactly one soldier for every 10 zombies, that will be enough to clear the city without leaving your base unprotected.

**D:** Since you’ll have one soldier for every six zombies, you’ll be sure to win, but because you’re sending in more than you need to, you’ll risk losing your base if zombies attack – and that would be a disaster.

**How to work it out:** The first thing you need to do is estimate the total number of zombies in the city. To do this, you need to account for the zombies hidden in the buildings by multiplying the counts for each city block by five (this gives 2,615, 3,160, 3,905, 2,105 and 1,535 as the estimated abundance in each block). Next, you need to work out the average number of zombies in a city block. This is done by adding up the counts in each block and dividing the total by the number of blocks sampled (in this case five). This gives an average of 2,664 per city block. You know that there are 142 similar-sized blocks in the city, so if you multiply the average number of zombies per block by this value (142) you will get the estimated abundance for the whole city. In this case, this is 378,288. Now you have the number of zombies, you can work out how many soldiers you need by dividing it by ten. This gives 37,828.8, or 37,829 if you round it up to the nearest whole number (since you can’t send in a fraction of a soldier!).

While the problems provided here are copyright of *Maths With Zombies*, if you are a teacher, you can use any of these problems for free in your classes – but please credit *Maths With Zombies * as the original source (e.g. Downloaded from *MathsWithZombies.wordpress.com*). You can download a PDF handout of this problem from here.

If you do use this problem in a class, please post a comment here to let me know how you used it and how it was received by your students. These problems cannot be used for any commercial purpose without express written permission.

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From the author of *For Those In Peril On The Sea*, a tale of post-apocalyptic survival in a world where zombie-like infected rule the land and all the last few human survivors can do is stay on their boats and try to survive. Now available in print and as a Kindle ebook. Click here or visit www.forthoseinperil.net to find out more. To download a preview of the first three chapters, click here.

To read the *Foreword Clarion Review * of *For Those In Peril On The Sea* (where it scored five stars out of five) click here.

I’m afraid you have a terrible miscalculation which would result in a failure to take back the city. In your explanation, you say that the numbers counted in each city block would have to be multiplied by five to account for the zombies that are hidden inside buildings. I’m afraid this is incorrect. By multiplying by 5, you are only accounting for those hidden zombies, and not for the original zombies that you counted out in the open. You must multiply the count in each sample block by 6 to obtain the correct number! By my calculations, you will actually need 45, 395 soldiers.

Dear Amelia,

Well spotted! In the problem it should have said that for every one zombie you see, there are four hidden from sight (giving a total of five) and not five as it originally said.

Thanks for pointing this out, and I’ve now corrected this in the post.

All the best,

Colin

Colin,

Just wanted to say thanks for the blog. I teach high school math in the U.S. and I am using some of your problems in my classes. My students love the excitement and novelty provided by your brilliant scenarios!

Amelia

Hi Amelia,

Thanks for letting me know that you use some of these problems in your classes, and it is great to hear your students love them. I need to get round to writing some at some point, but my day job keeps getting in the way!

Anyway, it’s great to know that I’m helping students get excited about doing maths.

All the best,

Colin