Hectoc is a Mental Calculation game, developed by Yusnier Viera in the 2010s. Each question is a sequence of 6 digits, each 1–9.
For example: 249384
In hectoc, the aim is to use these numbers in order to find the number 100. Here are two solutions to this example:
- 2 × (49 + 3 – 8/4) = 100
- (2 × 4 – 9 + 3)^(8/4) = 100
The operations allowed when solving hectocs are:
- putting adjacent numbers together, like 4 and 9 to make 49
- addition, subtraction, multiplication and division
- adding negative symbols
- powers (like the ‘^’ in the second example above)
- brackets to manage the correct order of operations
It is not allowed to rearrange numbers or use other symbols like square roots. You must use all the numbers.
You can read more about hectoc on its official website—including:
- How Yusnier invented the game while looking at the numbers printed on bus tickets in Cuba
- An app where you play this game for free, including easy levels
- The official rules
Hectoc has also appeared in some advanced international mental calculation competitions, such as the JMCWC and Memoriad. This article shares some of the strategies that can help you solve most hectocs like the above in less than one minute—sometimes in just a few seconds!
Almost all hectocs have solutions. Only 166 (0.03%) of all possible hectocs have no solution, according to an analysis by Mika Kuge (2019).
Summary of Strategies
Firstly, note that most hectocs have multiple solutions—for example 249384 above. When I practise hectoc with random numbers, I can usually find 4–5 solutions within a couple of minutes. So, you can try a variety of ideas to see which one first gives you a solution.
But there are millions of possible ways to combine numbers in any particular hectoc! And only a few of these would give the answer 100. So, if you just guess operations at random, it could be extremely slow to solve each hectoc.
A much better idea is to work backwards from 100. There are some ‘easy’ ways to make 100 from other numbers, such as:
- 10 × 10
- 99 + 1
- 25 × 4
This means you can select one of these ideas, like 10 × 10, and try to build the number 10 with the first digits, and then build 10 with the remaining (last) digits. If you can do this, then you have found a solution!
In this article, we will solve the hectoc 196264 by working backwards from 100 in a variety of ways—starting with the most useful ones that you should typically consider first. With this method, I can solve most hectocs in under 20 seconds. One of the most successful competitors internationally is Georgi Georgiev 🇧🇬 who scored 79/80 correct in 25 minutes in Bielefeld 2023.
Main Strategies
Strategy A: starting near 100
If you see a number close to 100 (or if you can build one easily), you can often add/subtract the other numbers to make exactly 100.
96 is close to 100, and this idea gives us solutions including:
- 1 + 96 + (2 × 6)/4
- (1 × 96) + 2 + 6 – 4
If we start with 26 × 4 (= 104, which is close to 100), we can also use the remaining digits to adjust the result to exactly 100:
- –(1 + 9 – 6) + (26 × 4)
This strategy is the simplest, and is the first thing I recommend to try if you see a ‘9’ within the first five digits of the question.
Strategy B: 100 = 10²
This is the second-most useful strategy, because it is easy to build 2 and 10. This is how I would solve this question using this strategy:
- Notice on the right that 6 – 4 gives 2. Can I make 10 using 1962?
- Not easily, but with enough creativity: 10 = 1 + (9 – 6)²
- The solution is (1 + (9 – 6)²)^(6 – 4)
Here is another way to apply the same strategy to solve the same problem:
- Notice on the left that 1 + 9 gives 10. Can I make 2 using 6264?
- Yes: 10 = (6 – 2) – (6 – 4) = 6 – 2 – 6 + 4
- The solution is (1 + 9)^(6 – 2 – 6 + 4)
Strategy C: 100 = 10 × 10
This is quite similar to using 100 = 10², except you need to build a 10 at the end, not a 2. Here are some solutions using this idea:
- (1 + 9) × (6 + 2 + 6 – 4)
- (1 + (9 – 6)²) × (6 + 4)
Notice that both of these solutions used ideas directly from my solutions from strategy B.
Strategy D: 100 = 20 × 5 = 5 × 20
Another common solution is to build 20 (or 5) and try to make the other factor to build 100. This is how I’d use this strategy:
- 19 is close to 20. Can I make 20 exactly using the first digits?
- Not as far as I can see… so we need to try something else!
- Can we make 20 using the last digits? Yes: 20 = 2 × (6 + 4)
- The remaining digits are 196. Can we make 5 from these? Yes: 5 = –1^9 + 6
- The solution is (–1^9 + 6) × 2 × (6 + 4)
Strategy E: 100 = 25 × 4 = 4 × 25
Similarly to Strategy D:
- (19 + 6) × (2 + 6 – 4)
- (19 + (6 × 2) – 6) × 4
Strategy F: add/subtract the first/last digit to change the target from 100
This method is technically Strategy A but the thinking is backwards. If you add (or subtract) the first (or last) number, then you would need to make a different number (not 100) with the other 5 digits. If that different number has lots of factors (like 99, 98, 96, 92, 91 or 102, 104, 105 or 108) then this might be easy. Here are some examples:
- Start with “1 +” so that with 96264 we would need to make 99. Often, this is possible using 99 = 11 × 9 or 99 = 9 × 11, but I don’t think that’s possible with this example.
- Finish with “+ 4” so that with 19626 we would need to make 96. This could be done with 4 × 24 or 12 × 8 or many other ideas. Here, we can do 100 = (19 – 6/2) × 6 + 4, and even 100 = 1 × (9/6) × 2^6 + 4
- Finish with “– 4” so that with 19626 we would need to make 104. This could be done with 8 × 13 or 26 × 4. For example, 100 = (1 + 9 – 6) × 26 – 4. This is similar to a solution found in Strategy A, but here our idea was to subtract the last 4, rather than to multiply by it.
Strategy G: 64 + 36
Both 64 and 36 can be created in a huge number of ways:
- 36 = 6² = 6 × 6 = 9 × 4 etc.
- 64 = 8² = 4³ = 2^6 = 8 × 8 etc.
Here, one solution is easy because the question literally ends in “64”:
- 100 = 1^9 × 6² + 64
I took part in a hectoc competition at Memoriad 2016, where one problem was 666666. I was very pleased with my solution of 100 = 6 × 6 + ((6 + 6)/6)^6, until someone pointed out the much simpler solution of 100 = (666 – 66)/6 🙃
Summary
When you see a new hectoc problem, first check the first and last digits to see which of the strategies are likely to work. For example, if the problem ends in ____73, you can probably use 100 = 10 × (7 + 3), but it is less likely that you can use the strategy 100 = 10².
Sometimes you need some creativity to use these strategies, but they will usually get you the answer—sometimes many different answers!
Otherwise, you can try more exotic combinations, but always work backwards from 100, rather than guessing combinations of operations.
Practice
You can practice using the official hectoc app designed by the game’s inventor, Yusnier—although it tends to give very simple questions to begin with.
The following formula generates hectoc questions on Google Sheets:
=100000 * RANDBETWEEN(1,9) + 10000 * RANDBETWEEN(1,9) + 1000 * RANDBETWEEN(1,9) + 100 * RANDBETWEEN(1,9) + 10 * RANDBETWEEN(1,9) + RANDBETWEEN(1,9)
(I might also make a simple practise tool and put it here someday.)
(If you know of another good hectoc practice tool, feel free to suggest it!)