Hidden Side of Board Games

Aug 8, 2022 | Paramus

This week we’re discussing our favorite multiplayer board games. It's no secret that these games are a wonderful way to socialize, exercise strategic thinking, and of course practice some basic arithmetic. But are there hidden secrets to discover? Read on! Oh, and since these are word problems of the week for which we give bonus stars for answers, our dear students reading this blog are ahead of their peers! Don't keep that a secret :-)

Lower Elementary: Risk: The Game of Global Domination is a strategy game with simple rules and complex results. There is lots of simple arithmetic, and as an added bonus, you’ll development geographical awareness and familiarity with the countries of the world in no time! Question: There are six continents on a Risk game board. If there are 12 troops in North America, 16 troops in South America, 18 troops in Europe, 22 troops in Asia, 24 troops in Africa, and 100 troops on the board in total, then how many troops are in Australia?

Upper Elementary: We spent hours as children playing Monopoly where keeping track of money is an must. What a good way to understand the exciting world of financial wheeling and dealing; with the added bonus of practicing arithmetic. Being banker is the best job, but no one should abdicate their fiduciary duties because alternative math is a hidden part of the game! Question: Each side of a Monopoly board has 11 spaces from one corner to the next. How many spaces are there around the whole board?

Middle School:  Scrabble is just an all round great game for mental agility and spatial awareness. Learn arcane words for the SAT, but more importantly, those crucial 102 double-letter words necessary to fit in tight spaces! The scoring requires addition, doubling, and tripling skills; and lots of mental math to maximize points! Question: In each set of Scrabble tiles, there are 42 vowels, 58 consonants, and 2 blanks. Nine of the tiles are A tiles. What fractional part of the tiles are not A tiles?

Algebra and Up: Clue is a game of logic and deduction using the process of elimination. If you think about it, we apply the same process algebraically when solving simultaneous equations! Question: In the game, there are 6 suspect cards, 6 tool cards, and 9 location cards. A combination of 1 suspect, 1 tool, and 1 location is selected at random at the beginning of the game and put in an envelope. What is the probability that the combination in the envelope is the Candlestick, the Library, and anyone but Colonel Mustard?

While you ponder those questions, have a look at the amazing mental math prowess of Professor Arthur Benjamin.

Professor Arthur Benjamin’s mathemagician skill is achievable with lots of training and constant practice. But unless it’s your profession, it’s impractical to devote that amount of time to developing and maintaining such prodigious computation skill. We’d rather devote our brain to examining, understanding, and strategizing solutions. However that requires number sense. Fortunately, we develop number sense by teaching simple mental math techniques at Mathnasium. Let’s see how we can compute these answers mentally with a minimum of effort.

Risk: We want our students to first strategize this two step solution before computing the answers (1) how many troops are not in Australia, then (2) removing that total from all troop must be the answer. When adding, we teach our students to “make tens”. Making tens requires knowing only these very few complements of ten: 1+9, 2+8, 3+7, 4+6, 5+5, and 0+10. Using mental math, we reorganize those troop numbers into: 12+18 = 2 tens and 2+8 = 3 tens, 16+24 = 3 tens and 4+6 = 4 tens, that makes 3 tens + 4 tens = 7 tens or 70, then 70+22=92. The number of troops in Australia must be 100–92. Since these numbers are close, we ask, “how far apart is it from 92 to 100?”  It’s 8 more. Great job! Regroup in other ways to check for correctness.

Monopoly: Playing Monopoly makes you realize that rolling a 10 jumps you to a similar space on the following side. That makes moving  by 10 easy! Then 11 means jumping by 10 and 1 space more, 12 is 10 and 2 spaces more. Then 9 must be jumping by 10 then back 1 space; and 8 is by 10 and 2 spaces back. We teach adding by 11, 12, 9, and 8 that way too – after we’ve ensured mastery of adding 10! Adding 11 is add 10 then 1 more, adding 9 is adding 10 then 1 less. Since there are 4 sides on the board, and jumping by 10 four times arrives back at the starting space, there must be 40 spaces. But why are there 11 spaces per side? It’s because each corner is shared by two sides. Since there are 4 sides, 4 corners are shared. So, 11×4 less the 4 shared corners = 44 – 4 = 40 spaces; and we’ve double checked for understanding!

Scrabble: The three step solution is (1) how many tiles are there in total, then (2) finding the non A tiles by removing the A tiles from that total, and finally (3) expressing that difference as a fraction. Using the making tens approach, we mentally total the tiles by grouping 58+2=60, 60+42=102 tiles. There are 9 A tiles, so the tiles that are not As must be 102-9. Since these numbers are far apart, we ask “How much is left?” Let’s use the complement to our Monopoly trick. Moving back 9 spaces is moving back 10 then forwards 1, or subtracting 10 then adding 1. 102 – 10 = 92 then add 1 to get 93. So the fraction of non-A tiles is 93/102. Let’s see if that is reducible. We can’t reduce by 2 – or any even number – because 93 is odd. To determine if we can reduce by 3 we use the divisibility by 3 rule; sum the digits and see if the sum is divisible by 3: 9+3=12, yes! 1+0+2=3, yes! Hence 31/34 fractional part of the tiles are not A tiles.

Clue: Probability of Candlestick is 1/6. Probability of Library is 1/9. Probability of anyone but Colonel Mustard is 5/6. Hence the combined probability = 1/6 × 1/9 × 5/6. The numerator is simply 5. We see 9 in the denominator and we have a mental math trick. Multiplying a number by 9 is multiplying by 10 then subtracting the number. So, 6×6=36, multiply by 10=360 subtract 36 is subtract 30 first to get 330, then subtract 6 to get 324 – notice how subtracting from front to back rather than backwards can be faster, and more importantly, we start with a good approximation of the answer. The answer is 5/324.

We hope we've revealed some hidden sides to board games. If not, then look under your game board the next time you're playing ;-)


Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), [email protected]
246 Main St. #A
Fort Lee, NJ 07024

Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, Palisades Park, North Bergen, West New York, and Fairview.

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