### The Lazy Gourmet, Part 2

This year hasn’t been the first that I discovered the joy of lazy recipes — I posted about cold brew coffee and no-knead bread a while back. However, I’ve made more additions to my slacker recipe collection since then.

- Breakfast is a glorious thing and pancakes are delicious. But I have to admit that even with a relatively simple recipe, there’s still the extra work of getting buttermilk, and the time it takes to babysit the pancakes, pouring them out in batches and watching them for the right time to flip. Because of this, I felt relieved when I started making dutch babies instead, which are basically one giant pancake you cook in the oven and slice up to share. Plus, they are a lot more fluffy than any pancake I’ve ever made, and taste great even with just a simple topping of lemon juice and berries.
- I own a pie bird, I’ll admit it. I think I’ve used it exactly once. And the apple pie that contained it was pretty delicious, but it was an awful lot of work. There’s the fact that you need two crusts for top and bottom (or if you’re masochist, a lattice crust on top). There’s the need for a starch in the filling so that it properly gels. There’s pre-baking the bottom crust. The egg wash. And, that bird. So from these over-engineered origins I recently switched to making galettes instead. This still has the crust, but only on the bottom, with the sides folded up on the edges to prevent leakage, or as someone in my family observed, it’s kind of an apple pizza. And, I was happy to find a recipe by Jacques Pépin that doesn’t need any fancy stuff. (Alton Brown’s apple pie recipe, by way of contrast, uses a half-dozen things I never have bought other than to make a pie.)
- Back in the day, I liked to make fajitas. These are pretty delicious, but require a ton of time spent in front of the stovetop, checking on the meat, peppers, onions, mushrooms and sauteeing them all in batches, only to have them get cold by the time you eat anyway. I switched this out for a shredded chicken recipe, which braises the meat in some simple ingredients and gives you a tasty sauce at the same time.

The recipes all use a food processor/stick blender, so I guess I have to recommend them to achieve optimal laziness. Here are the basic recipes.

- Dutch Baby
- Preheat the oven to 375 degrees F; bring 1/2 c milk and 2 eggs out of the fridge to come up to room temperature.
- Put 1 tbsp butter in a 8-10″ cast iron skillet, and leave 10 min in the oven. Melt 2 more tbsp of butter.
- Whiz 1/2 c flour, 3 tbsp sugar, 1/2 tsp salt in a food processor.
- Add eggs, milk, the 2 tbsp butter, 1/2 tsp vanilla to food processor and whiz about a minute, until homogeneous and frothy.
- Pour into pan, and bake 20 min without opening the oven.
- Serve with: berries and lemon wedges, or maple syrup.

- Apple Galette
- In a food processor, combine 1.5 flour, 1.5 tsp sugar, 1/4 tsp salt.
- Cut 1 stick (1/2 c) of cold butter into small pieces and put into processor. Prepare 1/3 c of vodka or rum, chilled with ice. (Or, ice water.)
- Process the dry ingredients + butter, drizzling the cold liquid into it, for 15 seconds or less. The dough should still be a little crumbly, but stick to itself when you squish a bit by hand.
- Form into a rough disk and refrigerate 1-24 hours. When ready to continue, take this out of the fridge to warm up a bit, to be rollable.
- Peel 4 apples. Slice: I use a gizmo like this to get slices that cook uniformly. Take half the slices and chop into small pieces (1/2 cm).
- Roll the dough. Avoid doing this in a hot area (i.e. right next to the oven), since you want the butter to remain solid. Aim for something like a 14″ diameter circle or a 12×14″ rectangle.
- Layer on the tasty things, leaving a 1″ border to fold later: add the chopped apples; 1 tbsp honey; the sliced apples; 2 tbsp sugar mixed with 1/2 tsp cinnamon; and 1 tbsp butter cut into small pieces. Fold up the sides.
- Bake 1 hour, or until browned. Let cool before slicing and serving.

- Shredded Chicken for Tacos
- In a dutch oven or heavy pot, briefly brown 1.5 lb of boneless chicken (thighs and/or breasts) in oil.
- Add a 14oz can of crushed tomatoes, 2 garlic cloves, 1 tsp each of oregano, cumin, and powdered onion, 4 chipotle peppers in adobo sauce, and salt+pepper to taste.
- Simmer, covered, about 45 minutes. If it looks dry, add some beer.
- Remove the chicken and shred with 2 forks.
- Puree the cooking liquid, e.g. with a food processor or stick blender and add back to the chicken.
- If you need to reheat, use a frying pan + oil, working in small batches.

Filed under: food, recipes | Leave a Comment

### Minimum-cost Flows

About a month ago, the Facebook Hacker Cup included a nice programming problem involving a certain kind of matching. The *min-cost bipartite matching problem* can be stated like this:

- You have a number P of people, and a number Q ≤ P of jobs. Each person is willing to do any of the jobs, but each person will charge a different price for each job. You are only allowed to hire each person to do at most one job, and you need every job to get done. What is the cheapest way to hire employees for all the jobs?

Just as a quick concrete example, let’s say we have P=3 people (Alice, Bob, Carl) and Q=2 jobs (Vacuuming, Window-washing).

- Alice charges $50 to do vacuuming and $40 to do window-washing.
- Bob charges $30 to do vacuuming and $20 to do window-washing.
- Carl charges $60 to do vacuuming and $30 to do window-washing.

In this case, the minimum cost to get all the jobs done is $60: pay Bob $30 for vacuuming, and pay Carl $30 for window-washing.

I had known about one way to solve this sort of problem, but I later learned another approach called *The Hungarian Algorithm*. It’s a very efficient approach, taking O(Q²P) time. In this post I’ll explain how it works, and in the next post, I will explain how it generalizes to an algorithm to the *min-cost max-flow* problem, specifically the *capacity-scaling successive shortest paths algorithm.*

## Basics

There are some preliminaries that I am going to build on here, which are generally covered in any introduction to flows:

- bipartite matching can be thought of as a flow problem, by attaching a source node
**s**to all people, directing edges from people to jobs, and attaching all jobs to a sink node**t** - given some flow in a graph, the
*residual graph*represents the unused capacity of the edges, together with added*reverse edges*that represent the cost that can be saved by decreasing the flow on that edge- in the min-
*cost*flow setting, a reverse edge’s cost is the negative of the original edge’s cost (i.e., cost savings)

- in the min-
- an
*augmenting path*is a path in the residual graph (using edges with nonzero capacity) from the source**s**to the sink**t**

The simpler *max-flow* problem, where we don’t care about costs, can be solved like this:

- As long as the residual graph has an augmenting path from
**s**to**t**, increase the flow on any augmenting path.

There are several ways to prove this algorithm’s correctness. Here is one way, which will be relevant later on:

Let F be some non-maximum flow, and X be any maximum flow. View the difference between X and F as a flow in F’s residual graph. By the

flow decomposition theorem, this difference can be written as a nonempty list of augmentings–tpaths plus a possibly empty list of cycles. Therefore, there is an augmentings–tpath, and we can keep improving F.

Let N be the number of nodes, and M be the number of edges. For unit capacities, this simple max-flow approach takes at most N iterations of a path-finding algorithm, which itself takes M steps, and so the total time complexity is O(MN).

Moreover, this simple algorithm works quite nicely when we add capacities to the edges (but still ignoring costs):

- Taking the
*shortest*path each time yields a maximum flow in O(MN) iterations; - taking the
*fattest*path (highest residual capacity) yields a maximum flow in O(M log K) iterations, where K is the maximum capacity.

## Greed and Costs

A *greedy* algorithm is usually one that just makes the cheapest decision at every point in time, without worrying about troubles that can come down the road from not planning ahead. Here’s the simplest greedy algorithm you could imagine for our matching problem now that costs are involved:

- Repeatedly do the following: out of all the people you haven’t hired and jobs you haven’t filled, pay the cheapest price you can to get one more job done.

However, this algorithm is not correct. In our example above, we’d first pay Bob $20 to do window-washing, since that’s the cheapest of all prices. After that, the cheapest price that doesn’t involve Bob or window-washing is to pay Alice $50 to vacuum. This total cost of $70 is not optimal.

Nonetheless, even once you add costs into the picture, there still remains a very elegant greedy algorithm that successfully solves min-cost flow:

- As long as the residual graph has an augmenting path from
**s**to**t**, increase the flow on any**minimum-cost augmenting path**.

So in our example, the second iteration would use the augmenting path [pay $30 for vacuuming by Bob, refund $20 from Bob for not window-washing, pay $30 for window-washing by Carl] which has cost $30-$20+$30 = $40, less than the direct path [pay $50 for vacuuming by Alice].

I was surprised the first time that I read this algorithm is correct! But, it has a very nice inductive proof. In the proof, the *value* of a flow is the amount of flow it sends from the source to the sink (so max-flow is really the problem of finding a flow with maximum value).

For simplicity, assume all capacities are unit. Let F be a minimum-cost value-

kflow. Let X be any minimum-cost value-(k+1)flow. View the difference between X and F as a flow in F’s residual graph. By the flow decomposition theorem, this difference can be written as a single augmentings–tpath P plus a possibly empty list of cycles.None of the cycles can have negative cost, because they could have been used to augment F into a cheaper value-

kflow. Therefore, augmenting F by P gives a value-(k+1)flow whose cost is at most X’s cost, and which is therefore a minimum-cost value-(k+1)flow.

Let’s assume unit capacities for now (which is true in the application to min-cost bipartite matching). This algorithm needs O(N) iterations. How can we find the minimum-cost augmenting paths? The residual graph has negative-cost reverse edges, but crucially, the above proof shows that it never has a negative-cost cycle. So the Floyd-Warshall algorithm in O(N³) time, or the Bellman-Ford algorithm in O(NM) time, can be used in each iteration. This gives a total time complexity like O(N²M).

We can do better!

## Using Your Full Potential

The Hungarian Algorithm, when viewed in the context of flows, lets us improve the above algorithm simply by switching to a better algorithm for shortest paths. Namely, Dijkstra’s algorithm. The only problem is those pesky negative edges; Dijkstra’s algorithm won’t work with those!

There is, however, a way to transform some graphs to avoid negative edges. The transformation is to use a *potential function*, which I’ll denote by **y**, since it’s like a dual solution for the associated linear program.

- A potential
**y**is a vector that assigns a real value to every node. - A
*valid*potential is one that satisfies, for every edge uv, y(v) ≤ cost(uv) + y(u). - The
*reduced cost*of edge uv is defined as reduced_cost(uv) := cost(uv) + y(u) – y(v);- so a potential is valid if all reduced costs are nonnegative.

Potentials are a way of transforming a graph in a superficial way:

- Let the
*reduced distance*from u to v be defined as the shortest path distance from u to v, along the reduced costs. - Then you can prove that the reduced distance from u to v doesn’t depend on the path:
- the reduced distance equals the normal distance from u to v, plus y(u), minus y(v).

Conversely, if we compute the reduced distances, we can easily recover the real distances. So as long as the algorithm can maintain a valid potential, the reduced costs are nonnegative, and it can use Dijkstra’s algorithm to find the shortest **s**–**t** path.

How can we maintain a valid potential? It turns out that not only are potentials useful for finding shortest paths, but also shortest paths are useful for computing potentials.

- In a graph with no negative cycles, define
**y**(v) for every node v to be the shortest-path distance from**s**to**v**. Then this**y**is a valid potential.

This is useful algorithmically because it lets us leapfrom from a valid potential, to shortest paths, to the augmenting path needed to augment the min-cost flow, to the next potential. Expressed this way, the Hungarian algorithm looks like this:

- Let
**y**be the all-zero vector. - Using
**y**as a valid potential, find a shortest**s**–**t**path**P**, stopping if none exists. - Redefine
**y**to be the shortest path distances from**s**. - Push a unit of flow along
**P**and update the residual graph. - Go back to step 2.

To prove that this actually works, we need to show that when we update the residual graph, **y** remains a valid potential.

This is because the edges

uvon any shortest path fromssatisfy satisfy d(s, u) + cost(uv) = d(s, v), which is a form ofcomplementary slackness.So adding the reverse edgevuwith cost(vu) = -cost(uv) will not violate the validity constraint y(u) ≤ cost(vu) + y(v) for the potential, as it holds with equality.

That’s it!

## Remarks

Some explanations of the Hungarian algorithm are slightly different in step 3, rephrasing it as the equivalent “Increase **y**(v) by the reduced distance from s to v, for each node v.”

Historically, the algorithm is based off of ideas of Kőnig and Egerváry, with Kuhn in 1955 being the first to consider a polynomial-time implementation. This algorithm is the oldest one that I know of where you get a substantial benefit from thinking both in the primal (the flow) and dual (the potential) at the same time. Similar ideas were used in 1965 by Edmonds to derive an algorithm for min-cost non-bipartite matching; more on that to come in the following post.

The algorithm above will work eventually for graphs with arbitrary edge capacities; but if the maximum capacity is K, the number of iterations might be something like K or larger. In the next post we’ll get that back down to polynomial-time by giving an algorithm that takes O(M log K) iterations.

Filed under: algorithms, graphs, programming | Leave a Comment

### ¾ million problems!

I wanted to post a short update on CS Circles, my project with the University of Waterloo CEMC to teach introductory programming online.

I’ve been checking the total number of problems solved on the site, and it has just surpassed 750000! I’m pretty sure I’ve never done 750000 of anything in my life except maybe eating grains of rice. More than 20000 users have solved the first few problems, while for the hardest problems, only slightly more than 500 students have solved them.

One of the most interesting things about CS Circles is how the “ask for help” feature has worked in practice. This feature is available on every programming problem, and allows students who are stuck to send in the current version of their code alongside a few sentences explaining where they are stuck. They can ask for help from their own teacher if they are registered on the site, otherwise they can ask the “CS Circles Assistant” which is me and my colleagues Sandy and Troy. The interesting thing is that more than half of the questions I get are already solved before I log in! I take the weekend shift so the questions I see have been sitting there about 24 hours. I remember using the Piazza system at university where we strived to answer every question within hours! I guess we should just be more lazy and most people will figure it out sooner or later, after they’ve taken the time to verbalize exactly what is the problem.

The other update is about the source code. CS Circles became open-source back in 2013, thanks to a SIGCSE special projects grant. Over the last couple of years, a team of educators in Serbia extended the project in two ways: first, they created a Serbian translation; second, they extended the underlying infrastructure to add new functionality. This includes look & feel improvements, an involved improvement to run multiple courses at once (in their case, multiple difficulty levels), and the ability to run time-limited and feedbackless quizzes. It was great just to know that the project was useful to them, and we also were able to publish a report on the changes and how the students liked it.

Are you interested in learning either Serbian or Python? Then you can check it out at http://imi.pmf.kg.ac.rs/imipython/

Filed under: CS Circles, education, language, open-source | Leave a Comment

### Google Ontario Tour

This holiday, I went back to Toronto to hang out with family and friends. It was a great vacation! But I also had to work for a couple of days. I took this occasion to check out the different Google locations in Toronto and Waterloo and work from each for one day.

The Waterloo office is very new; the people working there moved in November from another location across the street. PM Justin Trudeau visited it last week for the “official opening,” to much fanfare! It’s a fairly large office, with several hundred engineers and an assortment of other Googlers. I took a couple of pictures of things that I liked there. The above picture was near one of the microkitchens, it reminded me of hipstery cafes, although pretty deserted this time of year.

The second picture, below, I can only assume is a time machine. A pretty good way to get the scoop on the competition!The Toronto location is located right downtown. It is quite a bit smaller and consists almost entirely of sales and business staff. They are the first place I’ve ever visited that had outdoor minigolf on a balcony:

Just for the record, that’s a mini-golf hole consisting of a replica of the Prince’s Gate (of CNE fame), and if you squint hard, you can see a scuplture of a polar bear in the background. Moving inside, I found this in a microkitchen:

The Tim Hortons loses some authenticity points since it’s only called “Cafe & Bake Shop” in the US as far as I know. However, I have to give them credit for the box of maple sap! Finding that for sale anywhere is a new trend to me, but it was pretty delicious!

Filed under: food, google, toronto, travel, waterloo | Leave a Comment

### A Mathematician’s Apology

I just finished reading “A Mathematician’s Apology,” written by the number theorist G. H. Hardy in 1940. I sort of assumed I had already read it when I was younger, but this turns out to be completely false.

How do I know that I’ve never read it? Before opening it this time, I thought it was a book focusing on his relationship with the prodigy Ramanujan. (Was Hardy the one who gave him tuberculosis?) Nope, Ramanujan is barely mentioned at all.

Instead, the book is Hardy’s defense of mathematics. In brief, he laments that the kind of advanced mathematics he studied was not useful; and in the book he defends his life choice to spend all his professional time studying it.

Another major theme is that Hardy was past what he considered to be the mathematical prime of his life: he was age 62, although he would go on to live for another 7 years. He famously wrote,

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.

This, of course, is questionable, despite the examples Hardy gives in the book. Consider, for instance, László Babai’s apparent discovery this year of a quasi-polynomial-time algorithm for Graph Isomorphism, at the distinguished age of 65.

Beyond simply talking about perspectives on age, the book is very wistful and has a somewhat depressed perspective. I’ll admit people in inter-war Europe had reason to be troubled; the armed forces seemed to be so much more important than peaceful intellectual pursuits. It is too bad, though, that Hardy did not have the benefit of hindsight.

He wrote,

The general conclusion, surely, stands out plainly enough. If useful knowledge is, as we agreed provisionally to say, knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then

the great bulk of higher mathematics is useless. Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics — no one of them stands the test much better than another, and there is no real mathematician whose life can be justified on this ground. If this be the test, then Abel, Riemann, and Poincare wasted their lives; their contribution to human comfort was negligible, and the world would have been as happy a place without them.

This is pretty far from how things have turned out. Quantum mechanics and relativity have made possible things like transistors and GPS satellites. The theory of numbers and algebra led way to encryption which has had far-reaching effects on our daily life. Hardy actually admitted as much just a few pages earlier:

Time may change all this. No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the ‘highbrow’ applied mathematics will become ‘useful’ in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life.

So I don’t know if his conclusions of “everything is useless” is from a British sense of self-deprecation, or just griping from his yearning for youth.

I found the contrast even starker here:

Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

In a delicious twist of fate, Hardy’s job at Oxford put him just about 50 kilometers from Bletchley Park, where his compatriots such as Alan Turing and Bill Tutte were using combinatorics to reveal the secrets of the Nazis in World War 2. Admittedly, what they did was even beneath the “theory of numbers;” they used group theory, heuristics, and improvements to brute-force methods along with computer engineering. Even so, Hardy specifically disparaged pursuits of this ilk, calling out newspaper puzzles and chess problems as just having a mathematical “kick.”

All said, I still liked the book. Hardy talks at length about the aesthetics of mathematics and these discussions really struck a chord with me: once you’ve proven a mathematical statement, it stands true forever.

Now, I need to go (re-)learn about Gaussian integers in order to understand Hardy’s comments about the 2-squares theorem!

Filed under: books, math, reviews | 1 Comment

### BBB: Black Bean Burgers

A couple of years ago I got a subscription to Cook’s Illustrated, a fabulous food magazine. It has a few distinctive features:

**no ads!**this hugely increases my happiness in reading the magazine cover-to-cover- the recipes are written like science experiments
- the interiors are all black-and-white, with some photos replaced by schematics or pencil-sketch-style drawings (see right)

I’ve added quite a few delicious items from the magazine to my regular rotation of dishes, including carne deshebrada and wheat berry pancakes. This month’s issue had something entirely new for me.

The recipe is for a vegetarian burger. I’ve had veggie burgers before, but they always had the look, taste and texture of hockey pucks. To the contrary, these ones are pretty satisfying on all counts. Especially if you like your burger loaded with fixin’s, you can barely notice the difference in switching from beef burgers to these beany bad boys.

Here’s the outline of the recipe. I imagine you can use whatever herbs and spices are your favourite and it will turn out great no matter what.

- Drain
**two 15-ounce cans of black beans**. Let sit on paper towels to dry thoroughly. - Weigh
**one ounce of tortilla chips**by into a food processor. Pulverize. - Add the beans to the food processor and process until they have the texture of ground beef (approx 5 pulses).
- Beat
**two eggs**. Incorporate**2 tbsp flour**; this will be the glue to keep the burgers together. Add**herbs**(4 green onions and a half-bunch of chopped cilantro leaves) and**spices**(1 tsp cumin, 1/2 tsp coriander, 1/4 tsp each salt and pepper, 1 tsp sriracha, 2 minced garlic cloves). Stir. - Combine “glue” with bean-chips mixture and stir with a spoon until uniform.
*Important! Let the mixture rest, covered, in the fridge. Rest for at least 1 hour and at most 1 day.*- Form about 8 patties (or 12 to 16 sliders). Fry in a generous amount of vegetable oil at medium-high heat for 5 minutes per side.

We still have a couple left (pictured above). Just to make sure I don’t forget how to be a carnivore, I’ll be eating one topped with bacon slices for dinner.

Filed under: food, photos, recipes | 1 Comment

### Mexico

Better late than never, I wanted to post some memories from the trip we took to Mexico this summer for my brother’s wedding. Disclaimer: due to time zone changes, cerveza, or other factors, my memories of which thing happened on each day may have shifted.

Day 0: We got rebooked and had a slight delay but arrived in the afternoon to meet my family at the resort. It was a beautiful place. I played gigantic human-sized chess with my sister, who whomped me. At least there were free drinks to drown my sorrows. We went to the French restaurant on-site and had a delicious 3-course dinner.

Day 1: Had a great night’s sleep (no dream of being attacked by giant knights and bishops). We rendezvoused with siblings, siblings-in-law, and brother’s other friends at the various beaches and bars. We took taxis to Playa Del Carmen for a special dinner arranged by the newlyweds-to-be. Before returning home we checked out what I presume is one of the world’s largest souvenir stores, but managed to escape without purchasing anything.

Day 2: Ate a delicious breakfast (as usual); there were omelette stations and bottomless mimosas but I think on this day I composed some huevos rancheros. We started checking out the attractions off-site. Sara and I went to a particularly cool cenote named Río Secreto… it reminded me of past spelunking adventures but was much more chilled out. We skipped paying $100+ for the entire set of pictures but snapped a blurry bootleg of one image. We went to Cancún city afterwards; we got a little lost but then discovered some random outdoor concert/festival with lots of locals. It was a great adventure together, all in all. I tried out the churros. Verdict: they were delicious!

Day 3: My brother Brad’s wedding. Got my fancy clothes on with him and then we headed out to the bar for his last pretzel as a free man. The other groomsman Isaac was there to party in fine style (pictured: the fanciest bathroom I’ve ever been in). The wedding was on a beautiful beach, with beautiful people, and beautiful food. Brad and Claudia gave very heartfelt speeches and it was my honour to spend that day celebrating with them. There was an extremely delicious cocktail made with special Pisco (a national beverage for the bride). Then as it got dark we saw just how good the resort was at wedding planning 🙂 From what I recall we danced with pool noodles, glowing glasses, balloons, noise makers, and managed to stay until the closing of not one but two bars on the resort.

Day 4: Recovery. We played a lot of volleyball in the pool. Delicious mexican restaurant on the resort for dinner!

Day 5: Adventure. We rented two cars and drove out to Chichén Itzá. It was a very hot day but we had a good tour guide to keep us motivated. The driving was extremely peaceful as the highways were pretty empty. We learned that there were special acoustic effects built into the pyramids’ architecture, and that a lot of people “lost their heads” over competitive sports back in Aztec days. After escaping to some frosty beverages, we headed to Cenote Ik Kil; this cenote was a swimming hole set about 20 metres underground, with some fish patrolling the waters underneath us, and huge diving platforms overhead. Finally we headed to a delicious dinner of local Yucatan cuisine at Las Campanas, checking out Valladolid’s picturesque church before returning home.

Day 6: I went on a whale-shark expedition with my cousins’ significant others, Matt and Taka. (Sara has a fear of neckless animals so she decided to pass.) I managed to put on my contacts while on deck a boat speeding across the open ocean, which was exhilarating. We chased those whale-sharks (they are vegetarian sharks the size of whales) for a good couple of hours, stopping 4 or 5 times to get in the water with them. Each time we dived in we were disoriented, not just from being besnorkeled and finned, but also from the sheer scale of the creatures. They normally carried in tow an entourage of feeder fish to their sides, above and below. They were as large as a bus and moved just as fast but it looked effortless and so it was a strange feeling trying to chase them while swimming. It was an unforgettable experience! They served some ceviche with **hot** habanero peppers on-board, afterwards, and we did some more low-key snorkeling near a beautiful reef. We had one more dinner back on the resort and my family kindly endulged us to show them pictures of our trip to Iran, for which we repaid them in Persian sweets.

It was an amazing trip, both relaxing and full of unique celebrations. Congratulations again, Brad and Claudia! And thanks for bringing us all together on that awesome trip.

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