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Subject: concentration and calculation rss

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Trio is a simple game of calculation, designed to practice times-tables as well as addition and subtraction. It is aimed at young children, but can be enjoyable for the competitive geeks as well.

In the box, you will find 49 square tiles, each of which has a digit written inside a coloured frame. Frame colour and digit do not correspond to each other. The tiles are thick cardstock and have rounded edges. There are also 50 blue disks on which with the numbers 1 to 50 are printed in white.

Setup consists of randomly laying out the square tiles in a 7 by 7 grid and shuffling the disks face down.
The aim of the game is to get as many of the blue disks as possible. At the start of each round, one of the blue disks is revealed simultaneously to all players. They then try to find three square tiles which are adjacent in a horizontal, vertical or diagonal line such that multiplying two of the digits and then adding or subtracting the third gives the number on the disks. Whoever is the first to find and point out a solution obtains the disk. A new round is then started.
If all players agree that a solution cannot be found, the disk is removed from the game. If two or more players find (and point out) a solution simultaneously, the disk is also removed from the game.

This game is about speed but not only about speed. Although it is vital to be able to simply 'see' whether a particular combination of digits gives a specific result, to be successful one also has to remember partial results and where they occur. Otherwise one will search areas of the board where it is very unlikely to find a fitting combination of digits. In the same direction, once one has found that no combination of 6 and 7 (to give an example) exists whilst looking for 39 (= 6*7-3), when 46 comes up, it is no longer necessary to look for 6*7+4. Thus the game has a strong memory aspect.
But neither speed nor memory will win you a game against an experienced opponent unless you are able to absolutely concentrate for the duration of the game. This opens up the area of 'trash-talk' (if your opponents allow it) and of course mistakes (deliberate and otherwise). The game itself has no rule punishing announcing combinations which do not give the desired result, but it might need one. After all, the opponents need time to check whether your announced combination is true or not, and you could use that time to look for real solutions.

Some of the choices by the designer seem to need an explanation. Why take a 7 by 7 grid and not a 6 by 6 or 8 by 8? Why stop at 50 on the round disks? And why have the distribution of square tiles there is (namely 5 times '1', '7' and '8', 6 times '2','3','4','5', and '6', 4 times '9') and not a more equal one?
We can find some answers by running a couple of 1000 simulations on a computer. By doing so, we find that most targets (the numbers on the round disks) are reachable (i.e. there are combinations that yield the required number) in well over 99% of all games. The exceptions are 40 (98.3%), 42 (98.4%), 45 (98.3%), 48 (94.2%), 49 (98.7%) and 50 (97.7%). (Note that these numbers are all composite.) Extending the target range then gives 89.7% at 54 and 81.6% at 56. Since not occurring in more than 10% of all games is really not acceptable, 50 seems like a good place to stop (for a 7 by 7 grid).
Increasing the grid size to 8 by 8 would yield 99% at 48, 93.8% at 56, 90.2% at 63 and 57% at 72. So, increasing the number of square tiles by almost a third (from 49 to 64) would increase the acceptable range of targets to 62. Moreover, it is much easier to find lower numbers (e.g. on average there would be 24 combinations to achieve 15 compared to 18 combinations on the 7 by 7 grid), so who gets the lower numbers becomes more luck driven.
Decreasing the grid size to 6 by 6 leads to the target 40 being reachable in only 92.8% of all games and 48 in only 87.4% of all games. Decreasing the target range to 1 to 39 excludes the need to be able to handle 'big numbers' (like 6*7, 5*9, etc.) and leads to a much poorer game experience.
So a 7 by 7 grid seems like a good choice, if one wants to encourage use of all digits. (In a later article I might give more detailed analysis. Similarly, if anyone is interested in the C code with which I obtained these numbers, let me know.)
Finally, what about the uneven distribution? Evening it out (say 5 times '2' and 5 times '9' instead of the original) leads to slightly higher percentages for the 'problem target' 48 which then would be reachable in 95.9% of all games. It would make small target numbers only slightly harder to obtain. However, it could lead to 'clumping' in some games where the '8's and '9's are all close together and it becomes very hard to achieve some numbers. But on the whole, I would think that this change would improve the game slightly.

I already talked about the problem of deliberate mistakes.
The second problem is, in fact, how to set up the board. If one of the players does the set-up, he (or she) may have a significant advantage, because he might have already acquired and memorized knowledge about important subcombinations (like 6,7 next to each other).
Lastly, it can be difficult to reveal the next target, so that everyone sees it at once.

Variants and Handicaps
The basic game can be varied according to the preferences of the players. For example one may introduce brackets (allowing the formation of e.g. 6*(7+1)=48) or require four-number combinations (6*7 + 2*3 = 48).
Also, one can introduce a crude handicap system for uneven players. At the very basic level, one can force the better player to play looking at the grid from the wrong side. Increasing the handicap can be achieved by requiring the better player to find more than one combination yielding the target (beware that some targets might become impossible to achieve twice!). Alternatively, the better player may only use a 6 by 6 subgrid, or may not use a particular digit (say '5') at all.

Trio is a challenging and addictive game (yes, only for sad persons reading the 'geek) that is great for children learning their times-tables. Since it only takes about 10 minutes to play (two experienced players) one can have a game just after breakfast to wake up and again after dinner before one shuts down one's brain in front of the telly. Given its simplicity and short duration this is an excellent game for all those times 'in between'! The only sad news is that it is long out of print. There are copies on (German) e-bay going every now and then for around €4 plus shipping. It is well worth grabbing one of them - particularly if you have kids who still have to learn their times-tables!
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lotus dweller
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Good review of a very good game.
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