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I'm an English learner from Japan. Today I have questions not about the English language but about arithmetic, which I have found in a homework website for grade four kids. Here are the questions.

Each letter in each question stands for a 1-digit number. In each question, no two letters may stand for the same number. Two separate problems are unrelated. Find a value for each letter. (The sign * means 'multiplied by').

(1) ***=SNORE (Use the numbers: 6, 2, 0, 8, 4, 1, 3, and 9)

(2) ***=BRAND (Use the numbers: 1, 2, 4, 9, 3, 7, 8, and 5)

(3) ***=BULLET (Use the numbers: 1, 8, 6, 0, 5, 9, 2, and 4)

I have got the answers by using an almost try-and-error method. But the method takes really a lot of time:nearly 3 hours for each. Does anyone know the smartest way to solve this kind of alphametic quiz?

paco

I'm an English learner from Japan. Today I have questions not about the English language but about arithmetic, which I have found in a homework website for grade four kids. Here are the questions.

Each letter in each question stands for a 1-digit number. In each question, no two letters may stand for the same number. Two separate problems are unrelated. Find a value for each letter. (The sign * means 'multiplied by').

(1) ***=SNORE (Use the numbers: 6, 2, 0, 8, 4, 1, 3, and 9)

(2) ***=BRAND (Use the numbers: 1, 2, 4, 9, 3, 7, 8, and 5)

(3) ***=BULLET (Use the numbers: 1, 8, 6, 0, 5, 9, 2, and 4)

I have got the answers by using an almost try-and-error method. But the method takes really a lot of time:nearly 3 hours for each. Does anyone know the smartest way to solve this kind of alphametic quiz?

paco

Comments

if someone presented me with this problem, i would immediately implement this algorithm (brute force) with a simple computer program. it's really the simplest systematic way of doing this.

if you used logical heuristics instead, it may be frustrating, because it isn't systematic. after 9 hours, you probably thought of some good ideas. here are some:

look at the first one: (1) ***=SNORE. T must be either 1, 2, or 3. if T is larger than 3, then the answer needs to be 6 digits and instead of 5. if T is 0, then the answer would be less than 5 digits. you can also notice that either E is 0 or Y is 1 (since *** = E in the one's place). this only narrows it down a little, but it does help. you could also try this:

(1) ***=SNORE

(*** + *** + E) * (*** + *** + Y) = *** + *** + *** + *** + E

*** + (***+***)*1000 + (***+***+***)*100 + (***+***)*10 + *** = *** + *** + *** + *** + E

(***-S)*10000 + (***+***-N)*1000 + (***+***+***-O)*100 + (***+***-R)*10 + ***-E = 0

and here is the characteristic equation:

(***-S)*10000 + (***(R+I)-N)*1000 + (***(Y+E)+***-O)*100 + (***+***(E-1))*10 + ***(Y-1) = 0

which may not actually buy you anything, but doing it may help you see something else.

-c

bigchrisThe answers I got by the try-error method were as follows. If you made a program please check them.

TIE×TRY=SNORE ? 234×291=68094

CAN×SAY=BRAND ? 138×539=74382

DIE×BEST=BULLET ? 106×2694=285564

Have a nice Sunday

paco

paco2004Just for your next quiz:

This is not true:

BE * IT = LIE

32 * 16 = 512

Here for the one digit it is also: *** = E but neither E = 0 nor T = 1.

Also are you sure about the O(n^2) ? This would be an excellent algorithm, as it would solve even the hardest possible quiz, that is one that uses all ten digits, in only 100 operations, while a real brute-force algorithm would need 10! = 3628800.

cu

eagle2l84If your kid gets more homework of this kind use this link (requires Java on your PC):

http://www.geocities.com/CapeCanaveral/Campus/7705/metic/metic.html

Solutions for TIE * TRY = SNORE found this URL:

190 * 182 = 34580

180 * 143 = 25740

290 * 246 = 71340

230 * 279 = 64170

160 * 149 = 23840 +

270 * 239 = 64530

234 * 291 = 68094 +

295 * 217 = 64015

But only those marked (+) use only the allowed digits.

cu

eagle2l84anonymousanonymous