How many dice?

The figure below shows an “unfolded” view of a typical 6-sided die (d6), used everywhere from board games to casinos:

An "unfolded" view of a standard d6.

An “unfolded” view of a standard d6.

Each of the six faces is uniquely labeled with the integers 1 through 6.  But that’s not all; note that the values on opposite faces always sum to 7.  This is a standard arrangement applied more generally to other types of dice as well.  My set of Platonic solid dice– with n sides for n \in {4,6,8,12,20}— all have this same property.  Let us call such an n-sided die standard if all opposite faces sum to n+1.

Problem 1: How many different standard 6-sided dice are there?  That is, in how many ways can we label the faces with distinct integers 1 through 6, with opposite faces summing to 7?

Problem 2: What if we relax the standard (constant opposite sum) property?  That is, in how many ways can we label the faces with distinct integers 1 through 6, with no other restrictions?

Problem 3: Same as Problem 2, but for the other Platonic solids (d4, d8, d12, d20)?

I think these are great examples of problems for students that straddle whatever line there may be between mathematics and computer science.  Problems 1 and 2 can be solved as-is “by hand.”  (And they are interesting in part because the answers are perhaps surprisingly small numbers.)  But the usual mathematical machinery involved just counts the number of dice in each case; it’s an interesting extension as a programming problem to actually enumerate (i.e., list) them, display visual representations of them, etc.

Problem 3 is more challenging; the machinery is the same, but the larger numbers involved require some amount of automation in the housekeeping.

Now for what I think is a really hard problem motivating this post (Edit: although after a response from a reader in another forum with a very elegant solution, perhaps this is not as difficult as I thought it might be!):

Problem 4: Same as Problem 1, but for the other Platonic solids– that is, how many different standard (d4, d8, d12, d20) are there, with constant opposite sums?


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Distribution and variance in blackjack (Part 2)


This is a follow-up to the previous post describing the recently-developed algorithm to efficiently compute not just the expected value of a round of blackjack, but the entire probability distribution– and thus the variance– allowing the analysis of betting strategies.  This time, I want to look at some actual data.  But first, a digression motivated by some interesting questions about this analysis:

Why combinatorics?

I spend much of my time in my day job on modeling and simulation, where the usual objective is to estimate the expected value of some random variable, which is a function of the pseudo-randomly generated outcome of the simulation.  For example, what is the probability that the system of interest performs as desired (e.g., detects the target, tracks the target, destroys the target, etc.)?  In that case, we want the expected value of a {0,1} indicator random variable.  The usual approach is to run a simulation of that system many times (sometimes for horrifyingly small values of “many”), recording the number of successes and failures; the fraction of runs that were successful is a point estimate of the probability of success.

Usually, we do things this way because it’s both easier and faster than attempting to compute the exact desired expected value.  It’s easier because the simulation code is relatively simple to write and reason about, since it corresponds closely with our natural understanding of the process being simulated.  And it’s faster because exact computation involves integration over the probability distributions of all of the underlying sources of randomness in the simulation, which usually interact with each other in a prohibitively complex way.  We can afford to wait on the many simulation runs to achieve our desired estimation accuracy, because the exact computation, even if we could write the code to perform it, might take an astronomically longer time to execute.

But sometimes that integration complexity is manageable, and combinatorics is often the tool for managing that complexity.  This is true in the case of blackjack: there are simulations designed to estimate various metrics of the game, and there are so-called “combinatorial analyzers” (CAs) designed to compute some of these same metrics exactly.

My point in this rant is that these two approaches– simulation and CA– are not mutually exclusive, and in some cases it can be useful to combine them.  The underlying objective is the same for both: we want a sufficiently accurate estimate of metrics of interest, where “sufficiently accurate” depends on the particular metrics we are talking about, and on the particular questions we are trying to answer.  I will try to demonstrate what I mean by this in the following sections.

The setup

Using the blackjack rules mentioned in the previous post (6 decks, S17, DOA, DAS, SPL1, no surrender), let’s simulate play through 100,000 shoes, each dealt to 75% penetration, heads-up against the dealer.  Actually, let’s do this twice, once at each of the two endpoints of reasonable and feasible strategy complexity:

For the first simulation, the player uses fixed “basic” total-dependent zero-memory (TDZ) strategy, where:

  • Zero-memory means that strategy decisions are determined solely by the player’s current hand and dealer up card;
  • Total-dependent means that strategy decisions are dependent only on the player’s current hand total (hard or soft count), not on the composition of that total; and
  • Fixed means that the TDZ strategy is computed once, up-front, for a full 6-deck shoe, but then that same strategy is applied for every round throughout each shoe as it is depleted.

In other words, the basic TDZ player represents the “minimum effort” in terms of playing strategy complexity.  Then, at the other extreme, let’s do what is, as far as I know, the best that can be achieved by a player today, assuming that he (illegally) brings a laptop to the table: play through 100,000 shoes again, but this time, using “optimal” composition-dependent zero-memory (CDZ-) strategy, where:

  • Composition-dependent means that strategy is allowed to vary with the composition of the player’s current hand; and
  • “Optimal” means that the CDZ- strategy is re-computed prior to every round throughout each shoe.

(The qualification of “optimal” is due to the minus sign in the CDZ- notation, which reflects the conjecture, now proven, that this strategy does not always yield the maximum possible overall expected value (EV) among all composition-dependent zero-memory strategies.  The subtlety has to do with pair-splitting, and is worth a post in itself.  So even without fully optimal “post-split” playing strategy, we are trading some EV for feasible computation time… but it’s worth noting that, at least for these rule variations, that average cost in EV is approximately 0.0002% of unit wager.)

The cut-card effect

Now for the “combining simulation and CA”: for each simulated round of play, instead of just playing out the hand (or hands, if splitting a pair) and recording the net win/loss outcome, let’s also compute and record the exact pre-round probability distribution of the overall outcome.  The result is roughly 4.3 million data points for each of the two playing strategies, corresponding to roughly 43 rounds per 100,000 shoes.

The following figure shows one interesting view of the resulting data: for each playing strategy, what is the distribution of expected return as a function of the number of rounds dealt into the shoe?  The gray curves indicate 5% quantiles, ranging from minimum to maximum EV, and the red and blue curves indicate the mean for basic TDZ and optimal CDZ- strategy, respectively.

Expected value of a unit wager vs. round, for fixed basic TDZ strategy (left) and CDZ- strategy optimized for the current depleted shoe (right).

Distribution of expected value of a unit wager vs. round, for fixed basic TDZ strategy (left) and CDZ- strategy optimized for the current depleted shoe (right).  Five percent quantiles, from minimum to maximum, are shown in gray, with the mean in red/blue.

There are several interesting things happening here.  Let’s zoom in on just the red and blue curves indicating the mean EV per round:

Estimated expected value of a unit wager vs. round, for fixed basic TDZ strategy (red) and optimal CDZ- strategy (blue).

Estimated expected value of a unit wager vs. round, for fixed basic TDZ strategy (red) and optimal CDZ- strategy (blue).

First, the red curve is nearly constant through most of the initial rounds of the shoe.  Actually, for every round that we are guaranteed to reach (i.e., not running out of cards before reaching the “cut-card” at 75% penetration), the true EV can be shown to be exactly constant.

Second, that extreme dip near the end of the shoe is illustration of the cut-card effect: if we manage to play an abnormally large number of rounds in a shoe, it’s because there were relatively few cards dealt per round, which means those rounds were rich in tens, which means that the remainder of the shoe is poor in tens, yielding a significantly lower EV.

Keep in mind that we can see this effect with less than 10 million total simulated rounds, with per-round EV estimates obtained from samples of at most 100,000 depleted shoe subsets!  This isn’t so surprising when we consider that each of those sample subsets indicates not just the net outcome of the round, but the exact expected outcome of the round, whose variance is much smaller… most of the time.  For the red TDZ and blue CDZ- strategies shown in the figure, there are actually three curves each, showing not just the estimated mean but also the (plus/minus) estimated standard deviation.  For the later rounds where the cut-card effect means that we have fewer than 100,000 sampled rounds, that standard deviation becomes large enough that we can see the uncertainty more clearly.


So far, none of this is really new; we have been able to efficiently compute basic and optimal strategy expected return for some time.  The interesting new data is the corresponding pre-deal variance (derived from the exact probability distribution).  The following figure shows an initial look at this, as a scatter plot of variance vs. EV for each round, with an overlaid smoothed histogram to more clearly show the density:

Variance vs. expected value of a unit wager, for fixed basic TDZ strategy (left) and optimal CDZ- strategy (right).

Variance vs. expected value of a unit wager, for fixed basic TDZ strategy (left) and optimal CDZ- strategy (right).

It seems interesting that the correlation sense is roughly reversed between basic and optimal strategy; that is, for the basic strategy player, higher EV generally means lower variance, while for the optimal strategy player with the laptop, the reverse is true.  I can’t say I anticipated this behavior, and upon first and second thoughts I still don’t have an intuitive explanation for what’s going on.


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Distribution and variance in blackjack


Analysis of casino blackjack has been a recurring passion project of mine for nearly two decades now.  I have been back at it again for the last couple of months, this time working on computing the distribution (and thus also the variance) of the outcome of a round.  This post summarizes the initial results of that effort.  For reference up front, all updated source code is available here, with pre-compiled binaries for Windows in the usual location here.

Up to this point, the focus has been on accurate and efficient calculation of exact expected value (EV) of the outcome of a round, for arbitrary shoe compositions and playing strategies, most recently including index play using a specified card counting system.  This is sufficient for evaluating playing efficiency: that is, how close to “perfect play” (in the EV-maximizing sense) can be achieved solely by varying playing strategy?

But advantage players do not just vary playing strategy, they also– and arguably more importantly– have a betting strategy, wagering more when they perceive the shoe to be favorable, wagering less or not at all when the shoe is unfavorable.  So a natural next step is to evaluate such betting strategies… but to do so requires not just expected value, but also the variance of the outcome of each round.  This post describes the software updates for computing not just the variance, but the entire probability distribution of possible outcomes of a round of blackjack.

Rules of the game

For consistency in all discussion, examples, and figures, I will assume the following setup: 6 decks dealt to 75% penetration, dealer stands on soft 17, doubling down is allowed on any two cards including after splitting pairs, no surrender… and pairs may be split only once (i.e., to a maximum of two hands).  Note that this is almost exactly the same setup as the earlier analysis of card counting playing efficiency, with the exception of not re-splitting pairs: although we can efficiently compute exact expected values even when re-splitting pairs is allowed, computing the distribution in that case appears to be much harder.

Specifying (vs. optimizing) strategy

The distribution of outcomes of a round depends on not only the rule variations as described above, but also the playing strategy.  This is specified using the same interface as the original interface for computing expected value, which looks like this:

virtual int BJStrategy::getOption(const BJHand & hand, int upCard,
                    bool doubleDown, bool split, bool surrender);

The method parameters specify the “zero memory” information available to the player: the cards in the current hand, the dealer’s up card, and whether doubling down, splitting, or surrender is allowed in the current situation.  The return value indicates the action the player should take: whether to stand, hit, double down, split, or surrender.  For example, the following implementation realizes the simple– but poorly performing– “mimic the dealer” strategy:

virtual int getOption(const BJHand & hand, int upCard,
        bool doubleDown, bool split, bool surrender) {
    if (hand.getCount() < 17) {
        return BJ_HIT;
    } else {
        return BJ_STAND;

When computing EV, instead of returning an explicit action, we can also return the code BJ_MAX_VALUE, meaning, “Take whatever action maximizes EV in this situation.”  For example, the default base class implementation of BJStrategy::getOption() always returns BJ_MAX_VALUE, meaning, “Compute optimal composition-dependent zero memory (CDZ-) strategy, maximizing overall EV for the current shoe.”

However, handling this special return value requires evaluating all possible subsequent hands that may result (i.e., from hitting, doubling down, splitting, etc.).  When only computing EV, we can do this efficiently, but for computing the distribution, an explicit specification of playing strategy is required.

The figure below shows a comparison of the resulting distribution of outcomes for these two example strategies.

Probability distribution of player outcomes of a unit wager on a single round from a full shoe, using "mimic the dealer" strategy (red) vs. optimal strategy (blue).

Probability distribution of player outcomes of a unit wager on a single round from a full shoe, using “mimic the dealer” strategy (red) vs. optimal strategy (blue).

It is perhaps not obvious from this figure just how much worse “mimic the dealer” performs, yielding a house edge of 5.68% of initial wager; compare this with the house edge of just 0.457% for optimal CDZ- strategy.

Algorithm details

Blackjack is hard (i.e., interesting) because of splitting pairs.  It is manageably hard in the case of expected value, because expected value is linear: that is, if X_1 and X_2 are random variables indicating the outcome of the two “halves” of a split pair, then the expected value of the overall outcome, E[X_1+X_2], is simply the sum of the individual expected values, E[X_1]+E[X_2].  Better yet, when both halves of the split are resolved using the same playing strategy, those individual expected values are equal, so that we need only compute 2E[X_1].  (When pairs may be split and re-split, things get slightly more complicated, but the idea still applies.)

Variance, on the other hand, is not linear– at least when the summands are correlated, which they certainly are in the case of blackjack split hands.  So my first thought was to simply try brute force, recursively visiting all possible resolutions of a round.  Weighting each resulting outcome by its probability of occurrence, we can compute not just the variance, but the entire distribution of the outcome of the round.

This was unacceptably slow, taking about 5 minutes on my laptop to compute the distribution for CDZ- strategy from a full 6-deck shoe.  (Compare this with less than a tenth of a second to compute the CDZ- strategy itself and the corresponding overall EV.)  It was also not terribly accurate: the “direct” computation of overall EV provides a handy source of ground truth against which we can compare the “derived” overall EV using the distribution.  Because computing that distribution involved adding up over half a billion individual possible outcomes of a round, numerical error resulted in loss of nearly half of the digits of double precision.  A simple implementation of Kahan’s summation algorithm cleaned things up.

Two modifications to this initial approach resulted in a roughly 200-fold speedup, so that the current implementation now takes about 2.5 seconds to compute the distribution for the same full 6-deck optimal strategy.  First, the Steiner tree problem of efficiently calculating the probabilities of outcomes of the dealer’s hand can be split up into 10 individual trees, one for each possible up card.  This was an interesting trade-off: when you know you need all possible up cards for all possible player hands– as in the case of computing overall EV– there are savings to be gained by doing the computation for all 10 up cards at once.  But when computing the distribution, some player hands are only ever reached with some dealer up cards, so there is benefit in only doing the computation for those that you need… even if doing them separately actually makes some of that computation redundant.

Second, and most importantly, we don’t have to recursively evaluate every possible resolution of pair split hands.  The key observation is that (1) both halves of the split have the same set of possible hands– viewed as unordered subsets of cards– that may result; and (2) each possible selection of pairs of such resolved split hands may occur many times, but each with the same probability.  So we need only recursively traverse one half of any particular split hand, recording the number of times we visit each resolved hand subset.  Then we traverse the outer product of pairs of such hands, computing the overall outcome and “individual” probability of occurrence, multiplied by the number of orderings of cards that yield that pair of hands.

3:2 Blackjack isn’t always optimal

Finally, I found an interesting, uh, “feature,” in my original implementation of CDZ- strategy calculation.  In early testing of the distribution calculation, I noticed that the original direct computation of overall EV didn’t always agree with the EV derived from the computed distribution.  The problem turned out to be the “special” nature of a player blackjack, that is, drawing an initial hand of ten-ace, which pays 3:2 (when the dealer does not also have blackjack).

There are extreme cases where “standing” on blackjack is not actually optimal: for example, consider a depleted shoe with just a single ace and a bunch of tens.  If you are dealt blackjack, then you can do better than an EV of 1.5 from the blackjack payoff, by instead doubling down, guaranteeing a payoff of 2.0.  Although interesting, so far no big deal; the previous and current implementations both handle this situation correctly.

But now suppose that you are initially dealt a pair of tens, and optimal strategy is to split the pair, and you subsequently draw an ace to one of those split hands.  This hand does not pay 3:2 like a “normal” blackjack… but deciding what “zero memory” action to take may depend on when we correct the expected value of standing on ten-ace to its pre-split 3:2 payoff: in the previous implementation, this was done after the post-split EVs were computed, meaning that the effective strategy might be to double down on ten-ace after a split, but stand with the special 3:2 payoff on an “actual” blackjack.

Since this violates the zero memory constraint of CDZ-, the current updated implementation performs this correction before post-split EVs are computed, so that the ten-ace playing decision will be the same in both cases.  (Note that the strategy calculator also supports the more complex CDP1 and CDP strategies as well, the distinctions of which are for another discussion.)

What I found interesting was just how common this seemingly pathological situation is.  In a simulation of 100,000 shoes played to 75% penetration, roughly 14.5% of them involved depleted shoe compositions where optimal strategy after splitting tens and drawing an ace was to double down instead of stand.  As expected, these situations invariably occurred very close to the cut card, with depleted shoes still overly rich in tens.  (For one specific example, consider the shoe represented by the tuple (6, 6, 1, 5, 8, 2, 5, 9, 2, 34), indicating 6 aces, 6 twos, 34 tens, etc.)

Wrapping up, this post really just captures my notes on the assumptions and implementation details of the computation of the probability distribution of outcomes of a round of blackjack.  The next step is to look at some actual data, where one issue I want to focus on is a comparison of analysis approaches– combinatorial analysis (CA) and Monte Carlo simulation– their relative merits, and in particular, the benefits of combining the two approaches where appropriate.


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Serializing MATLAB data

Consider the following problem: given a value in the MATLAB programming language, can we serialize it into a sequence of bytes– suitable for, say, storage on disk– in a form that allows easy recovery of the exact original value?

Although I will eventually try to provide an actual solution, the primary motivation for this post is simply to point out some quirks and warts of the MATLAB language that make this problem surprisingly difficult to solve.

“Binary” serialization

Our problem requires a bit of clarification, since there are at least a couple of different reasonable use cases.  First, if we can work with a stream of arbitrary opaque bytes– for example, if we want to send and receive MATLAB data on a TCP socket connection– then there is actually a very simple and robust built-in solution… as long as we’re comfortable with undocumented functionality.  The function b=getByteStreamFromArray(v) converts a value to a uint8 array of bytes, and v=getArrayFromByteStream(b) converts back.  This works on pretty much all types of data I can think of to test, even Java- and user-defined class instances.

Text serialization

But what if we would like something human-readable (and thus potentially human-editable)?  That is, we would like a function similar to Python’s repr, that converts a value to a char string representation, so that eval(repr(v)) “equals” v.  (I say “‘equals'” because even testing such a function is hard to do in MATLAB.  I suppose the built-in function isequaln is the closest approximation to what we’re looking for, but it ignores type information, so that isequaln(int8(5), single(5)), for example.)

Without further ado, following is my attempt at such an implementation, to use as you wish:

function s = repr(v)
%REPR Return string representation of value such that eval(repr(v)) == v.
%   Class instances, NaN payloads, and function handle closures are not
%   supported.

    if isstruct(v)
        s = sprintf('cell2struct(%s, %s)', ...
            repr(struct2cell(v)), repr(fieldnames(v)));
    elseif isempty(v)
        sz = size(v);
        if isequal(sz, [0, 0])
            if isa(v, 'double')
                s = '[]';
            elseif ischar(v)
                s = '''''';
            elseif iscell(v)
                s = '{}';
                s = sprintf('%s([])', class(v));
        elseif isa(v, 'double')
            s = sprintf('zeros(%s)', mat2str(sz, 17));
        elseif iscell(v)
            s = sprintf('cell(%s)', mat2str(sz, 17));
            s = sprintf('%s(zeros(%s))', class(v), mat2str(sz, 17));
    elseif ~ismatrix(v)
        nd = ndims(v);
        s = sprintf('cat(%d, %s)', nd, strjoin(cellfun(@repr, ...
            squeeze(num2cell(v, 1:(nd - 1))).', ...
            'UniformOutput', false), ', '));
    elseif isnumeric(v)
        if ~isreal(v)
            s = sprintf('complex(%s, %s)', repr(real(v)), repr(imag(v)));
        elseif isa(v, 'double')
            s = strrep(repr_matrix(@arrayfun, ...
                @(x) regexprep(char(java.lang.Double.toString(x)), ...
                '\.0$', ''), v, '[%s]', '%s'), 'inity', '');
        elseif isfloat(v)
            s = strrep(repr_matrix(@arrayfun, ...
                @(x) regexprep(char(java.lang.Float.toString(x)), ...
                '\.0$', ''), v, '[%s]', 'single(%s)'), 'inity', '');
        elseif isa(v, 'uint64') || isa(v, 'int64')
            t = class(v);
            s = repr_matrix(@arrayfun, ...
                @(x) sprintf('%s(%s)', t, int2str(x)), v, '[%s]', '%s');
            s = mat2str(v, 'class');
    elseif islogical(v) || ischar(v)
        s = mat2str(v);
    elseif iscell(v)
        s = repr_matrix(@cellfun, @repr, v, '%s', '{%s}');
    elseif isa(v, 'function_handle')
        s = sprintf('str2func(''%s'')', func2str(v));
        error('Unsupported type.');

function s = repr_matrix(map, repr_scalar, v, format_matrix, format_class)
    s = strjoin(cellfun(@(row) strjoin(row, ', '), ...
        num2cell(map(repr_scalar, v, 'UniformOutput', false), 2).', ...
                                     'UniformOutput', false), '; ');
    if ~isscalar(v)
        s = sprintf(format_matrix, s);
    s = sprintf(format_class, s);

That felt like a lot of work… and that’s only supporting the “plain old data” types: struct and cell arrays, function handles, logical and character arrays, and the various floating-point and integer numeric types.  As the help indicates, Java and classdef instances are not supported.  A couple of other cases are only imperfectly handled as well, as we’ll see shortly.

Struct arrays

The code starts with struct arrays.  The tricky issue here is that struct arrays can not only be “empty” in the usual sense of having zero elements, but also– independently of whether they are empty– they can have no fields.  It turns out that the struct constructor, which would work fine for “normal” structures with one or more fields, has limited expressive power when it comes to field-less struct arrays: unless the size is 1×1 or 0x0, some additional concatenation or reshaping is required.  Fortunately, cell2struct handles all of these cases directly.

Multi-dimensional arrays

Next, after handling the tedious cases of empty arrays of various types, the ~ismatrix(v) test handles multi-dimensional arrays– that is, arrays with more than 2 dimensions.  I could have handled this with reshape instead, but I think this recursive concatenation approach does a better job of preserving the “visual shape” of the data.

In the process of testing this, I learned something interesting about multi-dimensional arrays: they can’t have trailing singleton dimensions!  That is, there are 1×1 arrays, and 2×1 arrays, even 1x2x3 and 2x1x3 arrays… but no matter how hard I try, I cannot construct an mxnx1 array, or an mxnxkx1 array, etc.  MATLAB seems to always “squeeze” trailing singleton dimensions automagically.


The isnumeric(v) section is what makes this problem almost comically complicated.  There are 10 different numeric types in MATLAB: double and single precision floating point, and signed and unsigned 8-, 16-, 32-, and 64-bit integers.  Serializing arrays of these types should be the job of the built-in function mat2str, which we do lean on here, but only for the shorter integer types, since it fails in several ways for the other numeric types.

First, the nit-picky stuff: I should emphasize that my goal is “round-trip” reproducibility; that is, after converting to string and back, we want the underlying bytes representing the numeric values to be unchanged.  Precision is one issue: for some reason, MATLAB’s default seems to be 15 decimal digits, which isn’t enough– by two— to accurately reproduce all double precision values.  Granted, this is an optional argument to mat2str, which effectively uses sprintf('%.17g',x) under its hood, but Java’s algorithm does a better job of limiting the number of digits that are actually needed for any given value.

Other reasons to bypass mat2str are that (1) for some reason it explicitly “erases” negative zero, and (2) it still doesn’t quite accurately handle complex numbers involving NaNalthough it has improved in recent releases.  Witness eval(mat2str(complex(0, nan))), for example.  (My implementation isn’t perfect here, either, though; there are multiple representations of NaN, but this function strips any payload.)

But MATLAB’s behavior with 64-bit integer types is the most interesting of all, I think.  Imagine things from the parser’s perspective: any numeric literal defaults to double precision, which, without a decimal point or fractional part, we can think of as “almost” an int54.  There is no separate syntax for integer literals; construction of “literal” values of the shorter (8-, 16-, and 32-bit) integer types effectively casts from that double-precision literal to the corresponding integer type.

But for uint64 and int64, this doesn’t work… and for a while (until around R2010a), it really didn’t work– there was no way to directly construct a 64-bit integer larger than 2^53, if it wasn’t a power of two!

This behavior has been improved somewhat since then, but at the expense of added complexity in the parser: the expression [u]int64(expr) is now a special case, as long as expr is an integer literal, with no arithmetic, imaginary part, etc.  Even so much as a unary plus will cause a fall back to the usual cast-from-double.  (It appears that Octave, at least as of version 4.0.3, has not yet worked this out.)

The effect on this serialization function is that we have to wrap that explicit uint64 or int64 construction around each individual integer scalar, instead of a single cast of the entire array expression as we can do with all of the other numeric types.

Function handles

Finally, function handles are also special.  First, they must be scalar (i.e., 1×1), most likely due to the language syntax ambiguity between array indexing and function application.  But function handles also can have workspace variables associated with them– usually when created anonymously– and although an existing function handle and its associated workspace can be inspected, there does not appear to be a way to create one from scratch in a single evaluatable expression.


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IBM Research Ponder This: August 2016 Puzzle

This month’s IBM Research puzzle is pretty interesting: it sets the stage by describing a seemingly much simpler problem, then asking for a solution to a more complicated variant.  But even that simpler problem is worth a closer look.

I won’t spoil the “real” problem here– actually, I won’t approach that problem at all.  Instead, let’s start at the very beginning:

Problem 1: You have 10 bags, each with 1000 gold coins.  Each coin should weigh exactly 10 grams, but one of the bags contains all counterfeit coins, weighing only 9 grams each.  You have a scale on which you can accurately measure the weight of any number of coins.  With just one weighing, how can you determine which bag contains the counterfeit coins?

This is a standard, job-interview-ish, “warm-up” version of the problem, with the following solution: arbitrarily number the bags 1 through 10, and take k coins from bag k, for a total of 55 coins.  If all of the coins were genuine, the total weight would be 550 grams; the measured deficit from this weight indicates the number of the counterfeit bag.

So far, so good.  The IBM Research version of the problem extends this in two ways (and eventually a third): first, what happens if more than one bag– or possibly none of the bags– might be counterfeit?  And second, what if there is a limited number n of coins available in each bag (where, for example, n=1000 above)?  From the IBM Research page:

If n \geq 1024 then one can identify the [subset of] counterfeit bags using a single measurement with an accurate weighing scale.  (How?)

That is, instead of knowing that exactly one bag is counterfeit, any of the 2^{10}=1024 possible subsets of bags may be counterfeit, and our objective is to determine which subset, with just a single weighing of coins carefully chosen from each bag.

The motivation for this post is that, although the statement above seems to telegraph the intended elegant solution to this “warm-up” version of the problem, (a) that intended solution can actually be implemented by requiring only n=512 coins in each bag, not 1024; and (b) even this stronger bound is not tight– that is, even before approaching the “real” problem asked in this month’s IBM Research puzzle, the following simpler variant is equally interesting:

Problem 2: What is the minimum value of n (i.e., the minimum number of coins in each bag), for which a single weighing is sufficient to determine the subset of bags containing counterfeit coins?

Hint: I ended up tackling this by writing code to solve smaller versions of the problem, then leaning on the OEIS to gain insight into the general solution.

[Edit: The puzzle statement on the IBM Research page has since been updated to reflect that the “nice”– but still not tight– bound is 512, not 1024.]


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Twisty lattice paths, all alike

I haven’t posted a puzzle in quite a while, so…

Every morning, I drive from my home at 1st and A to my work at 9th and J (8 blocks east and 9 blocks north), always heading either east or north so as not to backtrack… but I choose a random route each time, with all routes being equally likely.  How many turns should I expect to make?

And a follow-up: on busy city streets, left turns are typically much more time-consuming than right turns.  On average, how many left turns should I expect to make?

The first problem is presented in Zucker’s paper referenced below, and solved using induction on a couple of recurrence relations.  But it seems to me there is a much simpler solution, that more directly addresses the follow-up as well.


  • Zucker, M., Lattice Paths and Harmonic Means, The College Mathematics Journal, 47(2) March 2016, p. 121-124 [JSTOR]


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Recently my wife and I saw a double rainbow, shown below.  I’m a lousy photographer, but it’s there if you look closely, along the top of the picture:


There are plenty of existing sources that describe the mathematics of rainbows: briefly, rays of sunlight (entering the figure below from the right) strike airborne droplets of water, bending due to refraction, reflecting off the inside of the droplet, and finally exiting “back” toward an observer.


The motivation for this post is to describe the calculations involved in predicting where rainbows can be seen, but focusing on generalization: with just one equation, so to speak, we can let the sunlight bounce around a while inside the water droplet, and predict all of the different “higher-order” rainbows (double, tertiary, etc.) at once.  I think this is a nice “real” problem for students of differential calculus; its solution involves little more than the chain rule and some trigonometry.


The figure above shows the basic setup, where angle \alpha \in (0, \pi/2) parameterizes the location where a ray of sunlight, entering horizontally from the right, strikes the surface of the spherical droplet.  The angle \beta is the corresponding normal angle of refraction, related by Snell’s law:

n = \frac{n_{water}}{n_{air}} = \frac{\sin \alpha}{\sin \beta}

where n_{air}=1.00029 and n_{water}=1.333 are the indices of refraction for air and water, respectively.  (Let’s defer for the moment the problem that these indices actually vary slightly with wavelength.)

As a function of \alpha, at what angle does the light ray exit the picture?  Starting with the angle \alpha shown in the center of the droplet, and “reflecting around” the center counterclockwise– and including the final additional refraction upon exiting the droplet– the desired angle \theta is

\theta(\alpha) = \alpha + (k+1)(\pi - 2\beta) + \alpha

where– and this is the generalization– the integer k indicates how many times we want the ray to reflect inside the droplet (k=1 in the figure above).  The figure below shows how this exit angle \theta varies with \alpha, for various numbers of internal reflections.


The key observation is that for k>0, there is a critical minimum exit angle, in the neighborhood of which there is a concentration of light rays.  We can find this critical angle by minimizing \theta as a function of \alpha.  (Note that there is no critical angle when k=0, and so a rainbow requires at least one reflection inside the droplets.)

At this point, we have a fairly straightforward, if somewhat tedious, calculus problem: find \theta where \theta'(\alpha)=0.  It’s an exercise for the reader to show that this occurs at:

\alpha = \cos^{-1}\sqrt{\frac{n^2-1}{k(k+2)}}

\theta = 2\alpha + (k+1)(\pi - 2\beta)

Evaluating at k=1 yields an angle \theta of about 318 degrees.  This requires some interpretation: imagine a line between you and the Sun.  Then standing with the Sun behind you, a primary rainbow is visible along a cone making an angle of about 360-318=42 degrees with this line.

(Coming back now to the dependence of index of refraction on wavelength: although the variation is negligible for air, the index of refraction for water varies from about 1.33 for red light to about 1.34 for violet light, so that the rainbow is not just a band of brighter light, but a spread of colors from about 41.1 to 42.5 degrees… with the red light on the outside of the band.)

For k=2, the critical \theta is about 411 degrees, corresponding to a viewing angle of about 51 degrees… with two differences: first, the additional internal reflection means that the light strikes the bottom half of each droplet, making a complete “loop” around the inside of the droplet before exiting toward the observer.  Second, this opposite direction of reflection means that the bands of colors are reversed, with red on the inside of the band.

Finally, the angle for k=3 is about 498 degrees: such a tertiary rainbow may be seen facing the Sun (good luck with that), in a sort of “corona” around it at an angle of about 42 degrees.



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