Eliminate Possibilities

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• Chapter 3
• Eliminate Possibilities

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Emil Gordans tomcat and Pearls old nag were
not much use. I traded them for my new goat.

• Olive Hey, that sounds like a good
  cryptarithmetic problem. Lets see if we can
  solve it.
• G T O M
• P N A G
• E G O A T
• Each letter stands for a different digit.
• G 5 and A is odd.

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Make systematic lists.

0 1 2 3 4 5 6 7 8 9 G
G T O M P N A E 5
4
Generate possibilities.
In the thousands column, 5 P adds to
5. So P 0 and theres no carry to this
column or P 9 and theres a carry of 1 to this
column.
5
Eliminate possibilities by finding contradictions.
The thousands column adds to two digits, so P is
not 0.
P 0 or P 9
6
List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 E G P
G T O M P N A E 5 9 1
Therefore, P 9, theres a carry to this column,
and E 1.
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Generate possibilities.
In the tens column, O A adds to A. So O 0
and theres no carry to this column or O 9 and
theres a carry of 1 to this column.
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Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 E G P
G T O M P N A E 5 9 1
P 9, so O ? 9.
O 0 or O 9
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List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
Therefore O 0 (zero).
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Generate possibilities.
In the hundreds column, T N 10. So we have
these possibilities for T and N
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Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
T and N cant be 1, 5, or 9 because these numbers
are already used.
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Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
T cant be 2, 3, or 4 because M 5 T in the
ones column, and M cant be negative.
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List the remaining possibilities.
0 1 2 3 4 5 6 7 8 9 O E G P
G T O M P N A E 5 0 9 1
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Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
M cant be 1 because E 1.
G T O M P N A E 5 0 9 1
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Reread the problem.
G T O M
P N A G E G O A
T Each letter stands for a different digit. G
5 and A is odd.
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Eliminate possibilities by finding contradictions.
0 1 2 3 4 5 6 7 8 9 O E G P
M cant be 2 because then T and N would be the
unused odds, leaving none for A.
G T O M P N A E 5 0 9 1
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List the remaining possibility.
0 1 2 3 4 5 6 7 8 9 O E N M G T P
G T O M P N A E 5 8 0 3 9 2 1
T 8 N 2 M 3
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Finish the problem and check.
0 1 2 3 4 5 6 7 8 9 O E N M G A T P
G T O M P N A E 5 8 0 3 9 2 7 1
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Strategy Eliminate possibilities

• Generate possibilities.

• Eliminate possibilities by finding
  contradictions.
• List the remaining possibilities.
• Check them against the evidence. There may be
  more than one solution.
• Think about possibilities you might have missed.