Bubbling Cauldrons: Bring Out your Inner Alchemist!
For this activity, you will need the following materials: counters with numbers 1-30 on them, and 3 cauldron cutouts (you can print out this worksheet if you’d like).
Starting with just 2 cauldrons, place the counters into the cauldrons in ascending order – you can choose which cauldron each one goes in. However, there’s one catch: each of these numbers are highly potent alchemic ingredients, and under the right conditions, they can cause the cauldron to become unstable. If two numbers in one cauldron add up to a third number in that same cauldron, they bubble up and cause an explosion! This means that all the ingredients, or numbers, leave the cauldrons, and you must start all over again.
Our goal is to find the largest number we can place in either of the cauldrons without them exploding… do you think you’re up for this daunting task?
If you’re at all confused about the rules, you can watch the video below for a better visual:
Question 1: 2 Cauldrons
When I first approached this question, I didn’t have any sort of strategy. In fact, I didn’t even know when to stop, or if what I got stuck on was right. I just rearranged the numbers until I felt like I couldn’t get any further. Here’s what I found:
Ok, so hopefully 8 is the answer, I thought. Don’t worry – I’ve fact-checked this since then, and it is. However, it’s not all that clear in the beginning.
Question 2: 3 Cauldrons
Now, what if I raised the number of cauldrons to 3? Assuming I keep the same rules from the last problem, what is the largest number I can place in the three cauldrons?
Since I already found what I assumed to be the best distribution of numbers for 2 cauldrons, I decided to stick with that setup for the first two cauldrons in this problem. After that, the only option for 9 was the third cauldron.
Thinking more strategically, I realized that I could keep adding numbers consecutively to the third cauldron up until the number 18, because the smallest possible combination of numbers in that cauldron would be 9 + 10 = 19.
Then, I could go back to the first cauldron and place 19 there. Since 1 is in the first cauldron, I would then have to place 20 into the second. With 20 and 3 already in the second cauldron, the highest I could achieve in that cauldron would be 22.
After that, I was stuck again. 23 couldn’t fit in any of the three cauldrons. So, I decided, I’d found my answer.
Question 3: Start from 2
Let’s go back to 2 cauldrons again (because keeping track of numbers gets tricky with 3). What if, instead of starting at 1 with my counters, I started at 2? Now what’s the largest number I can reach?
Looks like 12 (we think). Ok, what about starting from 3?
I was able to get up to 17. By this point, you may have noticed my strategy – one that didn’t quite work for Question 1, but seems to work for starting the counters at 2 and 3 (and hopefully 4, 5, etc.). I’ll lay it out here in just three steps:
- Starting with your first number, put the smallest numbers (consecutively) in the first cauldron until you can’t go any further – or the two lowest numbers add up to the next consecutive number.
- Place the next string of numbers in the second cauldron, up to but not including the sum of the smallest two numbers in that cauldron. In this case, since 7 + 8 = 15, we can only place 7 – 14 in the second cauldron.
- Then, go back to the first and repeat the same process, but this time, place all the numbers up to the sum of the lowest number in the cauldron (your original starting number, or 3 for this example) and the new, “restarting” number (if you will), which is 15 here.
Challenge #1
Find an expression for the largest number we can place in two cauldrons starting with any positive whole number.
If you continue to use the strategy from Question 3, it’s pretty easy to find the largest number possible for the next few starting numbers:
- 2 yields up to 12
- 3 to 17
- 4 to 22
- 5 to 27
- 6 to 32
- etc…
You’ll notice that each of these numbers is 5 apart, so it’s easy to come up with a linear expression where the starting number is n. To get to 12 from 2, we simply multiply 2 by 5 and add 2 more. The same works for 3: 3 * 5 = 15 + 2 = 17. Therefore, the general expression for finding the largest number you can place in one of the cauldrons is 5n + 2.
Only one starting number strays from that pattern: 1, which goes to 8 instead of 7.
Challenge #2
Now, instead of the cauldron becoming unstable if 2 numbers add to a third number in the same cauldron, the cauldron only explodes if 3 numbers add to a fourth. Find an expression for the largest number we can place in two cauldrons starting with any positive whole number (again).
Using the same strategy as earlier, we find that if we start at 1, we get numbers up to 23.
- Starting at 2, we get up to 34
- From 3, we get up to 45
- 4 to 56
- 5 to 67
- 6 to 78
- and so on…
You can see from this clean pattern that if you raise the number you start with by 1, multiply that by 11, then add 1, you get the answer. That looks a little something like this: 11(n + 1) + 1, which can also be written as 11n + 12.
Congratulations! You’ve finished your official training at AIM’s very own Alchemist Academy! Now, it’s time for you to explore the mathematical world of chemicals and cauldrons all on your own. Good luck!