## Friday, April 29, 2016

### 4 Coins challenge

Recently, we were at lunch and the kids asked for a challenge to entertain themselves. I had just read this Numberplay column and had Lora Saarnio's Four Coins Problem in mind.

The basic rules are to choose values for coins in a new system so that any value from 1 cent to 10 cents can be made with a single coin or with two coins.

For our older two, this was a great on-the-go puzzle without pen and paper. Later, we got into some extension questions that required more careful organization of their observations and attempts.

Mostly, I want to report some of the extension questions that we found interesting. The key point: this problem is very accessible (J3 could also work on a simplified version) but it is ripe for a Notice & Wonder discussion that will reveal increasingly complicated and challenging extensions.

Note: we used names of neighboring countries, but none of the claims in these scenarios are true (as far as we know).

Number fussiness
In Burma, the president really likes the sequence 1, 2, 3. Is there a coin system we can suggest that includes coins with all three of those values?

No duplicates
We noticed that, for all the coin systems we created, there were always some values that were duplicated. What we mean is that there are two ways of using the coins to make that value. For example, in the coin system 1345, the following values can be made two ways:

4 = 4 = 1+3
5 = 5 = 1+4
6 = 3+3 = 1+5
8 = 4+4 = 3+5

Is there a coin system where all the values can be made only one way? If not, why not?

Two rival countries
Cambodia and Laos really love to compete with each other. As a result, they want coin systems where there aren't any shared coin values. For example, if Cambodia has a coin with value 5, then Laos can't have a coin value 5.

Is it possible? If not, what is the smallest overlap possible and how many possible coin systems achieve this smallest overlap?

Note: when we started this investigation, we had already designed a new system for Cambodia, so our starting question was whether there was a new system for Laos that wouldn't overlap this particular system for Cambodia, how much overlap was unavoidable, and then how to change both systems to minimize the overlap.

More values!
If we want to extend to amounts up to 20, how many coins are needed to have a compliant system (any value possible with only 1 or 2 coins)?

We haven't pursued this to a final answer, but the J's quickly recognized that, for any compliant coin system covering up to 10, they could create an 8 coin system that would cover up to 20. However, they also realized that four coins would be too few. This means that the minimum required to cover up to 20 is 5, 6, 7, or 8.

Coin triples
In the More Values! extension, we allow more than four coins in the system. The other way to relax the original constraints is to allow more than 2 coins to build a value. What can we achieve if we allow three coins at a time? Could we use fewer than four values in the system and still cover all amounts from 1 to 10? What about 1 to 20?

## Saturday, April 23, 2016

### Chocolate cake reference

This was Nigella Lawson's recipe, but that page currently doesn't include the ingredient amounts, so I'm reproducing it here. This was a test ahead of J3's birthday, so she helped. Since it seems successful, we will be making it again and I will include more pictures of the process.

Heat oven to 170 deg C
MIX:
• 3/4 cup + 1 tbsp flour
• 1/4 tsp salt
• 1/2 tsp baking soda

1/2 cup water (boiling)
6 tbsp cocoa powder
COMBINE into a thick paste and allow to cool (doesn't need to come fully to room temp)
2 tsp vanilla extract
ADD to cocoa paste and stir

2/3 cup olive oil
1 cup sugar
3 eggs
COMBINE in mixing bowl and mix on high (directions said 5 minutes, we mixed for 2-3)

COMBINE wet ingredients and mix thoroughly

COMBINE dry mix and wet just to incorporate together.

Use small amount of oil to grease a 9 inch baking pan.
Pour in batter and bake for 45 minutes (I rotated the cake every 10 minutes).

Another dessert reference to store, unfortunately dairy: Caramel Sauce

## Friday, April 22, 2016

### Ode to a bead string (a non-poem poem)

 from The Math Maniac

Whites and Reds
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5
From hand and eyes to brain
But also: a caterpillar, a snake,
dog collar and horse bridle
a gun
a gate or a fence
a telephone line
a part of a train
tiny eggs (maybe spiders?)
coins
jewelry (necklace, naturally, or earrings, bracelet, belt)
roller tracks for a made-up truck
beans cooking in a pan
and so much more!

While reviewing Dreambox Learning, I spent some time thinking about physical manipulatives compared to apps. Apps can certainly have nice features, but physical objects own my heart. I'm not even going to talk about the educational value (see this Hand2Mind note for a summary and further references). The thing we really like is the open-ended flexibility about how they can be used, both mathematically and for creative play.

As an example, the bead string is (maybe?) one of the more limited maipulatives because it is linear and one connected piece. We don't even have one. However, the ideas above were immediate ideas coming from observations of how my kids play and a quick brainstorming session with the little ones.
BTW, we find that a 6 to 9 year old can make any object into a gun, in case that idea didn't make sense to you.

 Favorite category of math manipulatives: food! In this case, freshly baked chocolate cake.

Returning to math, one of the cool things about this open play with math manipulatives is that it provides a lot of easy entry points into short math chats. These are fun, in themselves, and also reinforce the observation that math is accessible and all around us.

Manipulatives in Apps
While better than nothing, virtual versions of physical manipulatives always seem to fall short, with the following limitations:

• only one "correct" way to interact with them
• no possibility of combining
• can't take them apart
Perhaps there are also some benefits of virtual objects. In particular, they don't need to obey the law of conservation of mass. Can that, or other advantages, be used cleverly to make up for the disadvantages?

Let me know what you think and tell me where you've seen the best use of virtual manipulatives.

## Tuesday, April 19, 2016

### Tienanmen Attempts (Gord! festival continues)

We have been continuing the Gord! (BGG entry, Math Pickle site) games festival that started when we learned about Santorini. Today, the game we started learning is Tiananmen.

Game theme
While I have told the Js the name of the game and the designation of the sides as police/communist party and students, we haven't talked about the events of 1989. If you have a strong opinion about the game theme, feel free to let me know in the comments.

For now, this isn't our focus.

Creating a Board

This game is played with a go/baduk/weiqi set. While we recently got excited about that game, too, we haven't yet purchased a set. That means, once again, part of our efforts for this game are DIY/substituted versions.

Note: while the rules are very simple, we initially failed to understand how to make a legal move, thinking that we were able to play anywhere that was connected (including diagonally) to the batch of "stones" played by the previous player. That variation makes it far too easy for the students to win. In most of the pictures below, there will probably be a bunch of illegal positions.

As a first attempt, we played checkers vs dinosaurs (+mammoths) on a chess board:

For this version, we shrank the target in the center to one intersection. Also, you can see that we were playing on the spaces, not the intersections of the board:

In this version of the game, it was far too easy for the students to win.

For our second attempt, we used square tiles to recreate a full sized board. Unfortunately, this is how many yellow, green, blue, and red tiles we had:

That made for a nice little exercise: how many tiles are missing from the square?

Not so nice looking but we did fill in the rest of the board with other square tiles (the whites are from our 100 board, so this gives another way of quickly seeing how many were missing). As you can see, the TRIO set again came in as a central monument:

On the larger board, we used blue 1 cc cubes for the police and wooden 1 cubic inch cubes for the students. Eventually, we ran out of wooden cubes and had to substitute circular magnets and then colored tiles.

If you look carefully, you can see that we still failed to understand the rules at this point: playing on the squares rather than intersections and making illegal moves.

An epilogue and recommendation
Finally, we did figure out how to play. For those starting out, especially those without a go set, I strongly suggest playing first on the intersections from a chess board (9x9 grid) with a monument in the central intersection. This is a version that is easily accessible for young kids, but still has connections to the strategic thinking of the larger game.

Also, you don't have to lay out 324 tiles in a nice arrangement!

## Monday, April 18, 2016

### Castle overrun and storm the boat (creating games)

Playing Santorini with a make-shift set inspired the older two Js to create their own games. We test played J1's game, but haven't yet had a chance to test play J2's game.

# Castle Overrun

Equipment:
TRIO set. In particular, we used the 7x14 top, 5 short and 1 long magenta and 5 short and 1 long green straight connectors, 50+ 1x1x1 cubes. The long connectors are kings, the short connectors are soldiers. Key feature is that the cubes can stack on top of each other and on top of the soldier/king pieces.

Set-up:
Kings start in diagonally opposite corners with their soldiers one square away. Pre-placed cubes also start on positions (6, 3), (6, 5), (9, 3) and (9, 5).
You can see these details in the picture below (the extra chains of cubes were just on the board to give a tidy "resting" set-up when we weren't playing):

Turns
Turns alternate. On a player's turn, they move one of their pieces (the king or a soldier) one grid space (pieces can move diagonally). Then, they have an option of one of 3 possible actions:

• build a cube on an adjacent space (including diagonally adjacent). There is no limit to how high they can build. Think of this as building part of a wall or tower. A king can build two bricks on one turn.
• destroy the top cube on an adjacent space. This is tearing down a part of a wall.
• Attack an opponent's piece on an adjacent square. This involves putting a cube on top of that piece. The J's called this "pig-heading" and a piece with a cube on top is "pig-headed." If a soldier has 2 cubes on top, they get removed from the board. If a king gets 3 cubes on top, then it gets removed from the board.

Winning
First player to remove the opponent's king from the board is the winner.

Playability
There are some elements that suggest this game has possibilities, but the current rules quickly become a stalemate. First, we noticed that any attempt to pig-head an opponent is a half-suicide mission, since they can pig-head right back on their turn. There is one exception, but it involves the opponent accidentally walking into an obvious trap (and there's no reason why they would do that).

Also, the king can quickly build a fortress around himself before the enemy soldiers get close. Because the king can build two bricks on a turn, it would be impossible for the enemy soldiers to break through. In fact, this strategy is what gave us the idea for the name of the game.

Possible modifications
The main problem is that wall building is too fast. Perhaps we should allow the soldiers to break down two bricks each turn, but restrict all building to one brick at a time. This means that any fortress can, eventually, be broken down, if the siege army is unmolested.

# Storm the Boat

Equipment
Again, this is based on the TRIO set, but also requires 2 dice (choose your favorite sizes/shapes/markings).

Set-up
The straight connectors start in the same corner configuration as Castle Overrun:

Again, place one cube on each space (6, 3), (6, 5), (9, 3), (9, 5). Further, add bricks in a line from (1,2) to (4, 2) and (11, 6) to (14, 6).

There is another special piece in the center of the board, the boat:

Turns
Player rolls two dice and then moves a piece according to the values shown on the dice. They can move one piece for both values, or move two different pieces. Restrictions: movement is orthogonal (not diagonal) and pieces can not pass through a space occupied by another piece.

Pieces can move up one level on the boat. For example, if they are next to an open bottom magenta space, they can move onto the boat. If they are on the magenta level, they can move onto the blue seats.

After moving, the player pushes the boat one space toward their starting corner. This means that only one of the boat entry points is available at a time. This is like a moored boat rocking back and forth with the waves.

Winning
The first player to get three pieces onto the boat seats is the winner. They get a double win if their "king" is one of the pieces to get onto the boat.

Playability
This seems to be a solid racing game, similar to many other games for children. The boat mechanic adds a small twist. It seems playable and fun for kids, but lacks any deep strategy.

Possible modifications suggested have been:
(a) some way for the two teams to attack each other
(b) a way to release the boat from its anchor so that the movement is more extreme and/or erratic
(c) possibility for a team to attack the boat and sink it (perhaps to prevent an opponent from reaching the winning condition).

# Summary thoughts

Putting aside how good their games were, I was delighted with the ideas and enthusiasm both J1 and J2 had for creating their own games. While I had been hoping they would have been more pro-active about designing a DIY Santorini set the day before, it seems that exposing them to this gave them a boost of inspiration to try their own.

## Tuesday, April 12, 2016

### Powers of a permutation and Santorini review

Two unrelated topics today. Or ... I guess one could argue that everything in math is related, but I don't see direct connections myself. If you spot some, let me know in the comments.

# Rainbow permutations

We have a collection of crayons that can stack. Well, to be honest, mostly the kids break the tips off. The second most common use is to stick them on fingers as fancy fingernails. The fourth most common use is to actually draw or color with them.

J2 was engaged in the third most common use, stacking, when he noticed something new. He started with the colors stacked in rainbow order (R O Y G B Purple Pink). Then, he took 2 off the bottom and moved them to the top. Then, he took the bottom two and moved to the top and repeated. He noticed that, eventually, he got back to the starting order. Experimenting further, he tried the same process moving 3 from the bottom and repeating. Again, he eventually got back to the starting order.

Here is an example of the crayons stacked together:

In this case, he is moving blocks of 5:

After showing me, he suggested trying 4, 5, 6, then 1. He noticed a couple of things:

• moving 6 is like moving 1 backwards (from the top to the bottom).
• Similarly, 5 and 2 are related, 4 and 3
• 7 is prime, so maybe that is the reason the arrangement repeats
To test his hypothesis, he added another crayon (for 8) and tested. Again, he got back to the original arrangement. Hmm, doesn't need to be a prime!

Where should we take this first foray into group theory?

# Santorini

Gordon Hamilton of Mathpickle, one of our favorite game and puzzle resources, has a Kickstarter for the newest version of his game, Santorini.

I encourage you to check it out. (For what it is usual disclaimer applies, I'm not financially related to this game or producers in any way.)

For reference, this is what the previous version of the game looks like:

Our DIY board (version 1)
While this version of the game will have nice custom pieces, the underlying game can be played with almost any collection of stackable objects. For our introduction to the game, we tried using our TRIO blocks:

To explain what you're seeing:

• 1x1x1 cubes are used for building levels. I originally thought we would color coordinate (red for first level, pink second, etc), but J1 and J2 liked mixing up colors.
• 4 unit straight connectors for our builders: magenta versus green.
• Angle or arc connectors are radio antenae to serve as the fourth building level and block further building
• Flags and wings as boundaries of the 5x5 board
Reactions
Both older Js enjoyed the game play. We quickly played 5 or 6 times. This game clearly has a lot of depth. We will have to play a lot more to see what patterns we can identify and whether we can develop any opening strategies. They are also very eager to play with some god powers. Perfect way to spend the rest of the holiday this week!

The TRIO blocks have pros and cons for this game. One of the best features is that it makes the game set-up robust to tipping the board, knocking the pieces off, or otherwise unsettling the position. It was clear to see the sizes of the levels and also immediate to identify towers that had already gotten killed (with a fourth level antenna).

In our play, there were two drawbacks. First, the straight connectors are a bit hard to remove from their positions. Second, the higher towers (2 and 3 levels) sometimes obscured allowed diagonal moves in a way that the stacking tiles version didn't. I wonder if this will also be the case with the new version of the game as the building levels seem tall relative to the size of the builders.

Longer-term, I wonder about buying customized games vs playing with generic materials. It is easy for the kids to grab a box off the shelf and start playing. Somehow, I suspect they will be less likely to grab a building set and a notecard with rules.

More thoughts on DIY game versions

Before Santorini, I had gotten excited about making DIY versions of the GIPF project games. I was thinking:

1. these games will be fun to play, so great motivation to replicate them
2. interesting challenge to make our own triangular grid board (squares we already have in abundance)
3. good creativity prompt as we repurpose items
4. it would make the kids feel power over the game structure and rules, leading to exploration of variations and deeper thinking about structure.

However, the kids, particularly J1, were surprisingly unenthusiastic. With Santorini, again, they didn't take much initiative in developing our DIY version. However, I wonder if they will be more motivated to modify or replace the TRIO blocks board since they now see that the game is fun and have experienced some of the limitations of our current version.

On the other hand, maybe they'll just push me to buy the commercial version.

## Thursday, April 7, 2016

### Prodigy Math game (review)

Upfront,  I guess I'll reveal that I'm not being paid or otherwise sponsored to write this review. Once you read it, you'll be surprised that the idea ever occurred to you.

A good friend and fellow PROMYS supporter recently asked me to take a look at an online math game: Prodigy. The background sounded great: play, math, monsters, magic, adventure. What could be better? I spent a couple hours going through it and came away very disappointed.

Tracy Zager Big 3
Top 3 non-negotiable criteria, from this post:

1. Time pressure: none (Prodigy Game passes this hurdle)
2. Conceptual basis: none (PG fails this hurdle)
3. Mistakes handled productively: no (PG fails this hurdle)

So Prodigy Game score 1/3 against Tracy's Big 3. In case you don't want to bother to read her post, passing grade is 3/3.

The good
Clearly, the team has spent a lot of time and effort on development. There are some aspects that reflect this:

• Teacher back-end: this is where they put in their effort. Decent reports about student activity and teacher ability to pick focus questions (assignments) is nice. This functionality seemed on par with other edtech products I've used.
• Look and feel/animation: pretty good, clearly another area they prioritized. The standard is well below a popular commercial/non-education game, but I would say their work here is only slightly below the cutting edge in edtech products. It is much better than the huge crowd of flash animated drill and kill games.

Game theme: weak.

I think one reason I was so disappointed is that the premise starts out rather promising. We are going to explore a new world, gain experience, learn new spells, rather loot.

Unfortunately, each of these turns to disappointment. Most game play is driven by railroaded mini-storylines where we follow a guiding pointer along a linear path to retread locations we've seen before. There is limited opportunity or in-game reason to explore the world.

Gaining experience gives us extra hearts (capacity to take damage in magical duels), but this just makes the inserted mini-math quizzes longer. More experience feels like it makes the game less fun. Also, the way we gain experience is dueling random forest creatures, an activity that quickly seems pretty uninteresting and unmotivated (the forest creatures are just hanging out, they aren't bad/evil or doing anything wrong per se).

For each new spell, there is a cute animation showing the effects of that spell. However, they all have exactly the same in-game effects, so we never have a reason to care about the extra spells (though we do have to waste time and clicks choosing one each time).

Finally, they make two mistakes with the loot. First, we don't have a counter to keep track of how much gold we've collected, so it becomes hard to pay attention to that. Second, the vast majority of items you can collect or buy are only available for paid users.

The ugly
There is no math integration with game and theme. During each magic duel, we are forced to answer a math question as the hurdle to casting a spell successfully. That's right, doing math in this game is a cost that you have to pay!

This mechanic forces \math questions into the game play, but doesn't create any relationship relate to anything else about the game. The same format could be used for spelling, history, driver's ed, etc questions just as easily as math.

The math content itself is very weak. Unfortunately, the material included is pure drill, and even includes a lot that is really recall rather than skills practice. The questions had no context, no conceptual framework, were tedious. I literally found myself forming an active dislike of the material. Imagine what damage that could do for a kid who thinks this is a good representation of what math is!

Don't even get me started about the name "prodigy." Did the name bias me against the game from the start? Maybe, but I felt that my enthusiasm for games and the promised theme were stronger biases, so I still feel I gave the product a good chance.

Conclusion
This is a math "game" that I will never show to the three J's and I suggest you avoid it. Play Lure of the Labyrinth instead!

## Saturday, April 2, 2016

### Start of vacation math

Here in Thailand, the kids are now on summer vacation, so I've been exceptionally busy and found it hard to write blog posts. We've still been doing a lot of math, so I wanted to at least note down some of the activities.

# Sonobe Modular Origami

Credit for this one goes to another parent at J1's school. One day when school was still in session for J1, but J2 and J3 were already on vacation, we happened to see a group making sonobe modules at the elementary school. This is something my mother tried to introduce to the kids last year, but it didn't catch them at the time. Not sure what was different this time, but J2 was really intrigued.

Most of the work has been J2, but J3 also got into the act and folded about 9 units herself. I helped to sharpen the folds for 6 of them, which J2 and J3, working together, made into a cube.

Here are two of J2's stellated octahedra, made in between swimming sessions at the beach:

Video references we found useful for guiding our sonobe exploration:

Sonobe Cube: this is where we started.

Stellated Octahedron: shows how to make the basic module and how to assemble 12 into a stellated octahedron. This proved to be easier than I had anticipated, so J2 was able to do this all on his own.

Doubled tetrahedron: unfortunately, this doesn't give a sonobe module tetrahedron, but rather a 6 sided shape. We had actually made one of these ourselves when experimenting with 3 sonobe modules created from post-it notes. These are delightfully strong and feel really nice to toss from hand to hand.

Euler Characteristics
The model for the cube gives a triangulation for which it is particularly easy to count vertices, edges, and triangles. I talked about the Euler characteristic and we then did the counts for other sonobe models as well as other platonic solids to check. We also talked about the Euler characteristic for a cylinder (without end caps) and a torus.

Finally, I liked to describe our calculation as "0 dimensional things" - "1 dimensional things" + "2 dimensional things." This led J2 to ask what we would get if we included the space inside the shapes (extending to a "3 dimensional thing") and to get curious about 4 dimensional objects (which led to a lot of discussion, below).

Chirality
Sonobe modules come in two flavours, depending on which corners you fold into the center. All three of us (J2, me, grandpa) accidentally made the two kinds of sonobe modules without realizing it, only to struggle when we went to assemble a larger model. This started a brief conversation about chirality. While we didn't explore in depth, it did spark a later moment of recognition when Matt Parker mentioned the concept for mobius strips (see below).

Robustness
I was really surprised that the sonobe models we built are so robust. The completed shapes don't fall apart very easily (especially the doubled tetrahedron). Even better, for the models we have built so far, the sonobe modules themselves don't need to be perfect. This was crucial for the beginning origamists in our gang as it allows them to do almost all the work from start to finish.

It also made me think about my old idea of error in construction recipes (straight-edge and compass constructions as well as origami folding constructions). To illustrate this for my father (and J2 unintentionally) I showed them David Eisenbud/Numberphile's construction of the 17-gon. This, in turn, prompted J2 to get excited about straight-edge and compass constructions (below).

# Constructions

As mentioned above, J2 was excited about investigating classical construction problems. This is something he has seen a couple of times in the Abacus Math curriculum that P does with the kids. However, in that, we haven't generally been strict about the tools they use and the collection includes some standard triangles (45-45-90, 30-60-90) and measuring tools (rulers, protractors). Now, J2 was curious to see what could be done with just a straight-edge and compass.

For physical construction, our tool of choice has been the Safe-T compass:

Obviously, this doesn't quite correspond to the ancient limitations, but we ignore the ruled markings and the various holes are always within our margin of error on radius lengths anyway. Other than the obvious (for a parent) advantage, I think this compass style makes it easier for 2 people to help each other when drawing the circular arcs.

J2 has also been playing Euclid the Game. The virtual constructions have added two interesting points:

• strict restriction to what is constructed; he eyeballed a pair of equal lengths at one point, then wanted to talk about why he hadn't gotten credit for the construction (for angle bisection). This actually helped him identify that he needed the lengths to be equal; once he articulated that, he knew to use the compass to draw a circle.
• Chunking into advanced tools. For example, the game gives you a mid-point tool once you demonstrate the construction of a midpoint of a given line segment. This has helped him start to work on the coding concepts of functions/procedures and also gives him a new visual reference for the problem solving strategy "try to use what you already know."

# Fourth dimension

The big insight here is about the relationship between understanding four dimensional shapes and our own ability to perceive 2 and 3 dimensional shapes. Other than that tip, I'd direct you to these great resources:

Flatland: we don't actually have a copy, but I know it well and it is a foundation behind a lot of the discussions we had and videos we watched.

Matt Parker's Royal Institution talk: He covers a lot that isn't about four dimensional shapes, but I'm not going to point you to the relevant starting time because the whole talk is great!

XKCDhatGuy: This was actually one of the first videos J2 found and, hey, the kid does a pretty good job explaining. Don't, by any means, stop with this explanation, but it is a fine place to start.

Carl Sagan's Flatland: J2 liked it, but I still remember the old cartoon from the BU library that we would watch at PROMYS and wasn't so impressed.

4th dimension explained: [no comment]

Drawing 4th, 5th.. dimension:  Somehow, the simple points in this video really helped demystify higher dimensional shapes for J2. I think part of the point is that it encourages us to attend to aspects of the shapes that we can understand easily (for example, the number of vertices in an n-dimensional cube) rather than worrying about the macro shape.

Rotations of 4 dimensional shapes:  Similar message to the previous video

There is no 4th dimension: Nice, quick explanation living up to the high standards of other One Minute Physics videos. A very worthwhile series, in its own right.

# Cricket

This isn't mathematical, but I wanted to record this for posterity as well. Through random chance, J2 asked, "How do you play cricket?" At this point, he is still 2/3 English, having spent most of his life in London, so I guess he has a right to some information about this crazy pastime. For better or worse, though, that's not what he got. Our house rules:

1. batsman/batsmen pretend to stand in a traditional cricket batting pose, mime holding a bat
2. I stand a couple feet away, mime delivery of a cricket pitch (straight arm overhead throw)
3. we all (including spectators) run together into a giant heap, tickling each other and saying vaguely English-y, vaugely cricket-y things: "good show old chap," "ooh, looks like rain," "how do  you take your tea?" "you were almost to a century, young chap" etc
4. I break off the tickling and everyone stands in line, waiting for their points and an explanation of why points were awarded. Not spilling the tea is the most common reason for someone getting points, though boundary hits are also frequent. Note that points can be awarded in non-integer amounts (1/3, sqrt(2), and pi have all been awarded at some point).
5. Finally, I declare the match over with a result of "no result." As you can tell, we take our cricket seriously and are committed to the "test match" format.
Some important summary points:
• J2 is the player who has managed to accumulate the largest score in a single match. He was awarded "one hundred million one thousand" points by J3 for not spilling the tea.
• We are coining a new English-y phrase: "young old chaps." Usage example: when J3 is setting up a chasing game outside, she will point to J1 and J2 saying "these young old chaps are the runners," then point to me "and that old young chap is the monster."