Word Problems

Copyright ©2008 by Paul Niquette.  All rights reserved..
ot to be too fussy, but the word number in "The Secret Code" puzzle is itself something of a problem.  Whereas integer would be preferable, digit works best.  In the solution below, we have made that change, too.  Finally, "The Secret Code" puzzle did not specify that all of its digits were distinguished from one another.  Wanna bet, solvers worldwide are at this very moment taking that requirement for granted?  Only a scant few are making their assumptions explicit, which is our practice in Puzzles with a Purpose.
A woman wanted to log onto her work computer, but she had forgotten her ten-digit numerical password. However, she did remember ten clues: 
  1. The fifth number is the sum of the seventh and eighth.
  2. The tenth is the difference between seventh and eighth.
  3. The third number is the product of the seventh and eighth.
  4. The seventh number is twice the ratio of seventh to eighth. 
  5. The second is the seventh raised to the power of the eighth.
  6. The first is the difference between the second and tenth.
  7. The eighth number is the sixth less the fifth number.
  8. The ninth number equals the first less twice the fourth.
  9. The sixth number is one less than twice the fourth number. 
  10. The fourth is the eighth raised to the power of the eighth.

 
What were the ten distinct digits in the order given?
In solving Word Problems, it is a good idea to assign letters to all unknowns... 
  1. The fifth digit E is the sum of the seventh G and eighth H.
  2. The tenth J is the difference between seventh G and eighth H.
  3. The third digit C is the product of the seventh G and eighth H.
  4. The seventh digit G is twice the ratio of seventh G to eighth H. 
  5. The second B is the seventh G raised to the power of the eighth H.
  6. The first digit A is the difference between the second B and tenth J.
  7. The eighth digit H is the sixth F less the fifth digit E.
  8. The ninth digit I equals the first A less twice the fourth D.
  9. The sixth digit F is one less than twice the fourth digit D.
  10. The fourth D is the eighth H raised to the power of the eighth H.
Our ten 'clues' have been transformed into ten equations as follows:
  1. E = G + H
  2. J = G - H
  3. C = G H
  4. G = 2 G / H
  5. B = G H
  6. A = B - J
  7. H = F - E
  8. I = A - 2 D
  9. F = 2 D - 1 
  10. D = H H

efore diving in with algebraic manipulations, we might make a few observations...

  • Unlike the set-up in "The Secret Code" puzzle, combinations of unknowns are expressed here only in relation to one another not to any absolute numbers (quod vide 14, 10, 30).
  • The word difference denotes a subtraction (lines 2, 6, 7, 8, 9), but the outcome must always be positive, to make sense when pressing numerical code keys.
  • Exponents in line 5 and 10 must be decidedly restricted in the Land of Digits; indeed, with G not equal to H, the only values for B are either 8 or 9, (with H = 3 or 2, respectively).
  • However, H = 3 is ruled out by line 10, inasmuch as 33 = 27, which is not a single digit; accordingly H = 2, G = 3, and B = 9.
  • Knowing the values of H and G gives us solutions to lines 1 through 5 straight away, such that E = 5, J = 1, C = 6.  Line 4 is merely a confirmation that G = 3.
  • Therefore, per line 10, D = 4; per line 9, F = 7; per line 8, I = 0; and per line 7, F = 7.
The solution, then, is...
8-9-6-4-5-7-3-2-0-1

There is an unlimited number of Word Problems of this form.  You might baffle friends and family and strangers by creating your own.  Here's how...

  1. Write down a set of digits in some random -- certainly not mnemonic -- order.
  2. Make your job easy by giving them all symbols.
  3. Write a mathematical expression for each digit in terms of the others on the list.
  4. Translate each expression into words for presentation as 'clues'.
  5. Scramble the list of clues just to be mean or just to have fun.
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