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Grant's Review Corner: Volume 13

It's been a while since I've written one of these things.

I have had many, many, many years of experience with puzzles, both solving them and writing them. With this has come the ability to form opinions on puzzles and hopefully explain them with well thought-out prose to help other people form their own opinions; this is why there are 12 previous volumes of Grant's Review Corner. However, they are also very sparsely written: the last installment was over a year ago, and the one before that was 16 months prior! The primary reason is that I don't want to be inundated with requests to review things. However, formulating an opinion, dissecting it, and working out how to express it is a good exercise, so when I recently got a request to review an iOS app, I decided to do it.

Disclaimer: I was given a review copy of this app, normally $1.99, for free. While getting review copies of things is par for the course in the world of reviewing, this is a first for Grant's Review Corner.

Here is the email I received:

My name is Alexey Kuzmin! I've developed a small brain teaser app for IOS - Big Cat Lion (https://itunes.apple.com/us/app/big-cat-lion/id1116011744?ls=1&mt=8).

It's an alphametic app (a type of mathematical game consisting of a mathematical equation among unknown numbers, whose digits are represented by letters. The goal is to identify the value of each letter).

I'm writing to kindly ask you place a small review of my app in your blog. I could provide you a promocode to get my app for free to test it before writing review.

Please feel free to contact me if you have any question or suggestions.

Best regards,
Alexey

Alphametics, as mentioned above, are a puzzle in which letters represent digits in a simple mathematical equation. A slightly risque example I've seen in a few places is A+FAT=ASS (answer: 9+891=900). Almost all of them are in base 10, the base most humans use, so a computer can easily solve them (there are at most 10!=3,628,800 possibilities to check if you use brute force); if the base can get arbitrarily large, alphametics are NP-complete. The books Sideways Arithmetic from Wayside School and its sequel More Sideways Arithmetic from Wayside School, by Louis Sachar, feature, among other puzzles, alphametics with an amusing framing device of Mrs. Jewls teaching a very strange version of mathematics.

So how does our app Big Cat Lion stack up? Let's start with the logo:
It's styled after an alphametic. Cute. Unfortunately, it's not a valid alphametic; an eagle-eyed observer will notice that, owing to the commutative property of addition, any valid solution can have the values of G and T swapped and yield a second solution. Similarly, the values of B and C can be swapped. I'd suggest that Henry Dudeney's classic alphametic SEND+MORE=MONEY would make a better app title by virtue of having a unique solution, but in today's world which is saturated with freemium games, the connotations might actually make it a worse title.

The alphametics in Big Cat Lion are ranked in 4 difficulty levels, from Easy to Extreme. Let's take a look at one puzzle from each end of the spectrum.

 ADDLE
+VOICE
EAGLES

In the hundred thousands place, we have no digit plus no digit equals E. This should equal 0, but a number can't start with the digit 0. The logical conclusion, then, is that there's a carry in the ten thousands place, and that E equals 1. (This is an elementary deduction common to alphametics.) In the ten thousands place, we have A+V=A. It looks like V has to be 0, but again, a number can't start with 0, and furthermore we know the result must actually be A+10, rather than A, to create a carry for the hundred thousands place. Since V is a single digit, V=9 (and there's a carry from the thousands place). In the ones place, E+E=S; since E=1, S is easily shown to be 2.

 ADDL1
+9OIC1
1AGL12

The elementary deductions seem to end there. In the tens place, L+C=1 (or 11), and since we can't have multiple letters standing for the same digit, the possibilities are slightly constrained. We can make a table of these:

LC
38 [04567]
47 [03568]
56 [03478]
65 [03478]
74 [03568]
83 [04567]

I have included the unused digits next to each possibility for convenience. Note that Big Cat Lion has no capacity for making these tables; you'd have to use a sheet of paper or something else external to the app for this kind of thing, or else just try the possibilities one at a time and use a backtracking algorithm.

In the hundreds place, D+I=L. Actually, D+I+1=L (or L+10), because there's a carry from the tens place to account for in all of the possibilities. Thus, for each of the six possibilities above, we check the bracketed unused numbers for digits which sum to L-1 or L-1+10. We extend our table:

LCDI
3857 [046]*
3875 [046]*
4703 [568]
4730 [568]
4758 [036]*
4785 [036]*
5604 [378]
5640 [378]
6578 [034]*
6587 [034]*
7406 [358]
7460 [358]
8307 [456]
8370 [456]

Now we move to the thousands place, in which D+O=G. In the asterisked possibilities above, there's a carry from the hundreds place to contend for, and as shown earlier there must be a carry from the thousands place to the ten thousands place, so we can check the bracketed unused digits in each possibility for candidate values of O and G which make the math check out. Our possibilities get drastically reduced:

LCDIOG
385740 [6]
387564 [0]
4703 [568]
4730 [568]
4758 [036]
4785 [036]
5604 [378]
5640 [378]
6578 [034]
658743 [0]
7406 [358]
7460 [358]
8307 [456]
8370 [456]

The final unused value must be assigned to A. But A cannot be 0, because ADDLE cannot start with a 0, leaving just one solution: 65531+94781=160312.

  ACTUAL
+SADDEST
 STUFFED

No easy starting points seem to be present in this puzzle. The hundred thousands place has A+A=T, and we cannot have a carry to the millions place, but we might have a carry from the ten thousands place, so these possibilities exist for A and T:

AT
12 [03456789]
13 [02456789]
24 [01356789]
25 [01346789]
36 [01245789]
37 [01245689]
48 [01235679]
49 [01235678]

Seeing no better place to continue, I work on the ones place, L+T=D.

ATLD
1235 [046789]
1246 [035789]
1257 [034689]
1268 [034579]
1279 [034568]
1280 [345679]
1325 [046789]
1347 [025689]
1358 [024679]
1369 [024578]
1370 [245689]
1392 [045678]
2415 [036789]
2437 [015689]
2459 [013678]
2460 [135789]
2471 [035689]
2493 [015678]
2516 [034789]
2538 [014679]
2549 [013678]
2561 [034789]
2583 [014679]
2594 [013678]
3617 [024589]
3628 [014579]
3640 [125789]
3651 [024789]
3684 [012579]
3695 [012478]
3718 [024569]
3729 [014568]
3741 [025689]
3752 [014689]
3785 [012469]
3796 [012458]
4819 [023567]
4820 [135679]
4831 [025679]
4853 [012679]
4875 [012369]
4897 [012356]
4910 [235678]
4921 [035678]
4932 [015678]
4965 [012378]
4976 [012358]

4987 [012356]

Is that. . . 48 lines? This is already getting unwieldy and unfun. I don't want to make this blog post longer than it has to be, so I'm going to cheat and have a computer solve this for me.
Well, that wasn't so hard.

Actually, would you believe that ACTUAL+SADDEST=STUFFED is marked as the easiest difficulty in Big Cat Lion, and ADDLE+VOICE=EAGLES is marked as the hardest difficulty? I have no clue how this app grades the difficulty of its puzzles, but it doesn't do so with much accuracy. I wouldn't consider either of those to be easy (not in comparison to the pedagogical ones like ELF+ELF=FOOL in the aforementioned Wayside School books; even HISFOOT+TUTUTOO=WHOKNOWS, one of the harder ones in those books, is still a fair bit easier than both of the above examples from Big Cat Lion).

Since I got the initial request to review Big Cat Lion, an update to the app has completely revamped the difficulty scale. Now the individual puzzles aren't marked as easy or extreme, but instead when you play a puzzle, you can choose one of 5 difficulties. The easiest difficulty ensures you never have to solve more than two digits (ACTUAL+SADDEST=STUFFED becomes 458U43+7411278=78UFF21); the hardest difficulty never gives you any digits at the outset. Call me stubborn, but if a puzzle in an app has a unique solution, and I'm being given the option to have additional unnecessary information revealed at the outset to make the puzzle easier, I'll never take it! It just feels like an insult to my pride. If I want to solve an easy alphametic, I want one that's naturally easy like ELF+ELF=FOOL, not one that's been made easy by giving away information that could theoretically already be figured out (like 458U43+7411278=78UFF21). It's the same reason I play all of Everett Kaser's games (such as Sherlock) with a handicap of 0. For casual puzzlers, though, this is definitely better than the difficulty system we had before. Speaking of which, there's still a carryover from the old system: previously, the puzzles were listed in alphabetical order by difficulty, and they're still in the same order now despite not being graded, rather than all of them being alphabetized together.

At least the interface is clean. Scrolling through the list of puzzles (according to the Big Cat Lion website, there are 2000) is a bit of a chore, since you have to just swipe your finger a lot; I would hope there could be a bar on the far right for faster scrolling. If you're looking for a specific puzzle, or for puzzles with a specific word or string of letters, there's a search bar to save you the trouble of scrolling. I'm a furry, so let's look for puzzles with the word FURRY.
Well, that was enlightening, I suppose. Now I need a furry artist to draw a furry shopping at a floating mall and a furry with a briefcase getting tackled by a football player.

The interface for solving puzzles is also clean:
You tap the letter in the puzzle you wish to modify, and then tap the digit you want to assign to that letter. If that digit is already assigned to a letter, it will move to the new letter; if that digit is assigned to the current letter, the digit will revert to being a letter again. Alas, there is no undo button, which is especially egregious given that the interface disappears if you fill in the alphametic without making it add up correctly:
You have to start the entire puzzle over from its initial blank state if you do this. Do not do this.

I would like to mention a minor presentational issue. Remember how I mentioned that scrolling through the 2000 puzzles is a chore? In the process of doing so, I found a few with misspelled words (such as AKSED, CORSS, FIREND, NUGGTS, SAPCE, and THER). This really hurts the app in terms of looking polished. In the process of scrolling through the puzzles, I also found that the title puzzle, BIG+CAT=LION, is also there. According to a computer search from that website I cheated with above, the puzzle has 168 solutions (a multiple of 4, as expected since any solution has 4 variations). At least the author had the foresight to have the program check the mathematical accuracy of your solution rather than comparing it to the solution it has in mind; this means that any one of the 168 solutions will be accepted.

Overall, I have a medium impression of Big Cat Lion. It's better than that book Timothy Parker wrote and I reviewed (and to think that this review was before the plagiarism scandal that caused him to be the most hated person in the world of puzzles), but it's very difficult to find puzzles that are naturally easy and pleasant (like SHEEP+CHESS=COFFEE, another alphametic that could inspire good furry artwork, perhaps featuring Dawn Bellwether), and difficult alphametics (like ACTUAL+SADDEST=STUFFED) can feel very tedious and samey. It's why I like alphametics occasionally, but not in huge doses. A bigger selection of handmade and/or handpicked alphametics (like SEND+MORE=MONEY or I+BB=ILL) wouldn't hurt. I can't bring myself to say the kinds of things about this app that I did about these apps, but I'm not overwhelmed with positivity, either. It's alphametics, and that alone should mostly determine whether you like it or not.

If you have any counterpoints to offer in the comments section, I invite you to. I'm sure the author of this app would appreciate the discussion, especially if it's more positive than my "meh". :)

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