# Division!

When I was young, my Mom used to buy crossword puzzle magazines at the grocery store. She would do the crossword and the cryptograms, and I would do the other stuff: anagrams, logic puzzles, and of course number puzzles. My favorite was the kind with a long division problem where the digits have been replaced by letters. After you figure out the cypher and list the letters for 0 through 9 in order, they spell out a word or phrase, just to demonstrate that you got it right. Here’s an example:

```         KID
COOK|LACKING
LONGD
OKALN
LGABG
LGCKK
LDA
```

This says that LACKING divided by COOK gives KID with remainder LDA. We also know that

• COOK times K equals LONGD
• COOK times I equals OOADN
• COOK times D equals LGCKK

There are also some simple number facts that will come in handy:

• The rightmost column of a subtraction is simple. Say, if 8 minus Q is 2, then Q must be 6. If it’s not the rightmost column, Q might be 5, because the 8 might have been borrowed from.
• Pay attention to the ones digit in multiplications.
• If XYZ times Z ends in Z, then Z is 0, 1, 5 or 6.
• If XYZ times P, Q, and R all end in Z, then Z is 0 or 5.
• If XYZ times Q ends in Z, and Z isn’t 0 or 5, then either
• Q is 1, or
• Q is 6 and Z is even

OK, let’s start with the simple stuff. In the units place of the second subtraction, `N - N = B`, so B is 0. In the hundreds place, `A - A = G`. Since G isn’t also 0, G must be nine and there’s borrowing (D is bigger than L)

Also, `K * COOK` ends in D, not in K, so K can’t be 0, 1, 5, or 6. On the other hand, `D * COOK` ends in K, even though D is obviously not 1. So D is 6 and K is even. Since K*K ends in 6 and K isn’t 6, K is 4.

Let’s fill in what we have so far.

```         4I6
COO4|LAC4IN9
LON96
O4ALN
OOA6N
L9A09
L9C44
L6A
```

Since `9 - 4 = A`, A is 5. Since `A - O = O` with no borrowing from the column to the left, O is 2. `L - 6 ends in 5`, so L is 1. A borrowed-from 5 minus C is 1, so C is 3. Now we have:

```         4I6
3224|1534IN9
12N96
2451N
2256N
19509
19344
165
```

Almost done now. `I - 6 = 1`, so I is 7, and a borrowed-from 3 minus N is 4, making N 8. The full division problem is

```         476
3224|1534789
12896
24518
22568
19509
19344
165
```

And the word is

```0123456789
```

Note that this took nothing beyond grade-school arithmetic.

Here’s another one to practice on. The first correct solution gets, um, bragging rights. Have fun, and show your work.

(Hint: Abgr gur havgf qvtvgf bs gur cebqhpgf.)

```       ELI
MRO|MODELB
RLLE
RBFL
RDAL
OIFB
ORLI
ORM
```

By the way, there are only 10!, or about 3.6 million possibilities, so a computer program that brute-force tried all of them would run in well under a second. That would be cheating.

### Mike Schilling

Mike has been a software engineer far longer than he would like to admit. He has strong opinions on baseball, software, science fiction, comedy, contract bridge, and European history, any of which he's willing to share with almost no prompting whatsoever.

1. Oooh boy. I’m going to be attempting this so I’d ask that the people who are better at this than I am use rot13 for their answers… and by that I mean “spell your numbers”, if you please.

Thanks.

2. Jason Tank

V unir gb fubj jbex, qba’g V? Lrrfu.

Bx, rirelguvat gvzrf R vf rirelguvat, fb R zhfg or bar.
R – R naq Y – Y naq fhpu rdhny mreb fb S vf mreb.

Abj, n srj vasreraprf…

E – Z vf mreb, fb E zhfg or obeebjrq sebz. E = Z + bar.
O – Q rdhnyf bar jvgubhg obeebjvat sebz gur yrsg (E – E vf fubjvat mreb) ohg gurer’f n mreb gb O’f evtug, fb Q + gjb = O.
N + V = gra.

Q – Y vf fubjvat O, naq jr xabj Q + gjb vf O. Fb, Q + gra – Y = Q + gjb, juvpu fbyirf nf Y = rvtug.

Ng gur obggbz, Mreb – rvtug = E. Jr nyernql xabj E vfa’g bar, gurersber vg zhfg or gjb. (Naq gurersber Z zhfg or guerr fvapr E + bar vf Z.)
(Ng guvf cbvag, N naq V zhfg or sbhe naq fvk, ohg jr’er abg fher bs gur beqre lrg.)
O – sbhe lvryqf guerr jvgubhg obeebjvat sebz gur mreb, fb O vf frira. Q vf gjb yrff, fb vg’f svir, juvpu yrnirf R nf avar.
Mreb zvahf N lvryqf V jvgubhg orvat obeebjrq sebz gur evtug, fb N zhfg or 6 naq V sbhe.

Dhvgr n sbezvqnoyr ceboyrz, vs lbh xabj jung V zrna. V’ir arire npghnyyl qbar bar bs gurfr orsber whfg abj. Pbby!

• Mike Schilling

We have a winner.

One quibble: You say “Bx, rirelguvat gvzrf R vf rirelguvat, fb R zhfg or bar.” Hfvat bayl gubfr snpgf, vg jbhyq or cbffvoyr gung B vf fvk naq R, Y, naq V ner nyy rira. Gubhtu gung’f rnfl rabhtu gb qvfcebir (Fvapr S vf mreb, jr abj unir gur shyy frg bs rira ahzoref, ohg S – N vf V, jvgu ab pneel gnxvat cynpr. Bbcf.)

• Lbh boivbhfyl tbg vg evtug fvapr lbh unir vg pbeerpg va gur raq, ohg vg’f abg R gvzrf rirelguvat gung’f rirelguvat, vg’f B. Naq lbh pna ehyr bhg B = 6 rnfvyl, nf Zvxr fubjf.

Also, more of these please. It’s was an excellent time-killer.

• Jason Tank

Lrnu, gung ovg nobhg “Bu, V unir gb fubj zl jbex?” Gung jnf zr nsgre V nyernql unq guerr be sbhe ahzoref, gelvat gb erzrzore ubj V tbg gurz. V zvkrq hc n srj guvatf. (Ohg V pbzcrgryl zvffrq gur fvk ceboyrz.)

• Mike Schilling

Itâ€™s was an excellent time-killer.

High praise indeed.

• It really is. You have no idea how easily I get bored.

• I did this one sitting in an utterly pointless meeting.

• Mike Schilling

I could recommend some more books ….

3. Aaaaaah, I got sent offsite and my not solving it yet has nothing to do with me being dumb.

• The old offsite excuse. Used that one a hundred times myself.

• Seriously though, it is pretty tough. Once you have the first four letters, it will go quickly, but those first four are tough. It really is fun though, so worth the effort.

4. kenB

Hey, my parents used to get me those Dell puzzle mags as well (I’m assuming it was Dell). But they didn’t do any puzzles, so I had the whole thing to myself.

I got this answer quickly, but I don’t think you’d entirely approve of my method:

Gur svefg fhogenpgvba fubjf gung S vf gur svefg yrggre naq gurer’f na EZ va frdhrapr. Ybbxvat ng gur bgure yrggref, gur jbeq zhfg or SBEZVQNOYR. Fcbg-purpx gb irevsl.

• Mike Schilling

Ng yrnfg gurz jnf fbzr zngu. Tvira gur qvfpynvzre, V gubhtug lbh jrer tbvat gb fnl “V sbhaq nyy gur yrggref, naq srq gurz vagb na nantenz fbyire.”

• kenB

Perish the thought.

By the way, are you familiar with Erich Friedman? He’s a math professor at Stetson but he moonlights as a puzzle constructor — he’s got a bunch online here, and he has one or two brand new types of puzzle in every issue of Games Magazine. They fascinate me because it’s often not clear (to me) how to go about solving them — it seems like there’s enough pattern and order that there ought to be an algorithm, but all I can manage is to just start penciling in guesses, see what affects what, and hopefully come up with a few heuristics so that I’m not stuck with just trying every possibility.

• Mike Schilling

I’d never heard of him. I see what you mean about those puzzles. They’re frustrating.

5. S=mreb Guvf bar vf gur tvzzr.
B=bar orpnhfr R gvzrf B vf R, Y gvzrf B vf Y, naq V gvzrf B vf V.

Sebz gurer, V jrag gb thrff gur Y cyhf B jnf terngre guna 10, fb Z vf bar terngre guna E fb Z = E + 1

Ybbxvat ng gur arkg yvar qbja, V fnj ubj EOSY unq EQNY fhogenpgrq sebz vg naq svtherq bhg gung V + N = 10
Naq, gurersber, E + Y = 10
Jnf cyrnfrq jvgu zlfrys urer.

B + E + 1 = V
Fhofgvghgvba trgf hf gb E + 2 = V

Fb, hfvat fbzr zber fhofgvghgvba, V + N = E + Y
Juvpu zrnaf gung E + 2 + N = E + Y
Juvpu zrnafg gung N + 2 = Y

Ng guvf cbvag, V’z veevgngrq. V xabj rknpgyl ubj sne rirelguvat vf sebz rirelguvat ryfr.

Still digging.

• OHG JNVG! V ybbx ng gur gbc naq V frr gung E + Y rdhnyf 11! Vf guvf n oernxguebhtu?

E + 2 + N = 11
E + N = 9
Nyfb, N + 2 = Y

Abg ernyyl.

Ohg, ng yrnfg, V xabj jurer rirelguvat vf.

Jryy, yrg’f ybbx ng bhe yrggref.
0123456789
SB

Vf jung jr’ir cebira.
Jr xabj gung EZV vf va gurer.Jr xabj gung jurerire Z vf, Y unf gb or fbzrjurer gung nqqf hc gb 11. Naq tvira B, jr xabj gung EZV pna’g or 789.

23456789
N- Y-E ZV-

Tvira gung jr xabj gung V + N rdhnyf 10, jr pna ybbx ng EQNY orvat fhogenpgrq sebz EOSY naq xabj gung O vf 2 terngre guna Q (gb trg gur qvssrerapr bs 1 haqrearngu). Ner gurer n pbhcyr bs fcbgf va gurer gung ner gjb njnl?

Juvpu tvirf hf gur jbeq SBNQYOEZVR

Ng juvpu cbvag V tvir hc naq ernq fbzrbar ryfr’f nafjre.

• Oh, I see what I did.

I put the thing to the right when I should have put it to the left.

23456789
EZV-N-Y-

From there I could have… razza frazza.

• Jason Tank

Here’s the place where you’re screwing up. (This is not an answer, just showing what you wrote side by side.)

1fg pbzzrag: V + N = gra, N + V = E + Y. 2aq pbzzrag: E + Y = ryrira.

Gurer’f n qvfpercnapl gurer.

6. Mike Schilling

In case anyone’s interested, here’s how I did it.

Boivbhfyl, S vf 0.

Ol havgf qvtvgf va gur zhygvcyvpngvba, rvgure B vf 6 naq R, Y, V, nyy rira, be B vf 1. Vs gur sbezre, jr xabj nyy gur rira ahzoref (S, B, R, Y, V), ohg S – N = V, fb N vf rira gbb. Vzcbffvoyr, fb B vf 1.

Sebz Z – E orvat 0, jr xabj E + 1 = Z. Sebz n obeebjrq-sebz V – E = B(1), jr xabj E + 2 = V. Fb gur frdhrapr EZV vf fbzrjurer.

Fvapr R*E naq V*E obgu raq va Y, V naq R qvssre ol 5 naq E naq Y ner obgu rira. E > B(1), fb V + 5 = R.

Gur rira ahzoref ner S(0), E, Y, V(E+2), naq N (S-V).

N obeebjrq-sebz O zvahf Q vf 1, fb Q + 2 = O. O – V vf Z, fb V pna’g or 2, naq vg’f 5 fznyyre guna R, fb V vf 4 naq R vf 9. Guhf E vf 2 naq Z 3.

9 * 321 vf 2889, fb Y vf 8. 8 * 321 vf 2568, fb Q vf 5 naq N 6. O – 4 vf 3, fb O vf 7.

• The next time we get one of these, I’m totally going to get it right.