Lesson Logic problems - The solution to Problem 3

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<H2>The solution to Problem 3</H2>

Below &nbsp;(after the &nbsp;reminder)&nbsp; is the &nbsp;solution&nbsp; to the &nbsp;Problem 3&nbsp; of the lesson &nbsp;<A HREF = http://www.algebra.com/algebra/homework/word/misc/Some-logic-problems.lesson>Some logic problems</A>&nbsp; (the topic &nbsp;Miscellaneous word problems 
of the section &nbsp;Word problems&nbsp; in this site).

<BLOCKQUOTE>Problem 3 formulation (reminder)

Seven dwarfs decided that each of them will give a gift of at least &nbsp;7&nbsp; flowers to Snow White for her birthday.
When all the flowers were presented, &nbsp;Snow White said that the total number of the flowers is not more than &nbsp;50.
When the smallest dwarf heard it, &nbsp;he said immediately that he can identify exactly the number of flowers presented by each dwarf.
What is the number of flowers presented by each dwarf?</BLOCKQUOTE>

Solution

It follows from the problem statement that there are two major opportunities:
  &nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;each dwarf gave a gift of &nbsp;7&nbsp; flowers, &nbsp;totally &nbsp;49&nbsp; flowers, &nbsp;and
  &nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;one of dwarfs gave a gift of &nbsp;8&nbsp; flowers, &nbsp;while six others dwarfs  gave &nbsp;7&nbsp; flowers each, &nbsp;totally &nbsp;6*7+8 = 42+8 = 50&nbsp; flowers.

If it was the smallest dwarf who gave a gift of &nbsp;8&nbsp; flowers, &nbsp;then the only one option exists: &nbsp;each of the rest six dwarfs gave &nbsp;7&nbsp; flowers.
Therefore, &nbsp;in this case the smallest dwarf is in position to make his suggestion.

If, &nbsp;in opposite, &nbsp;the smallest dwarf gave a gift of &nbsp;7&nbsp; flowers, &nbsp;then two options exist:
  &nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;each of the rest six dwarfs gave a gift of &nbsp;7&nbsp; flowers, &nbsp;and
  &nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;one of the rest six dwarfs gave &nbsp;8&nbsp; flowers, &nbsp;while five others gave &nbsp;7&nbsp; flowers each.

In this case the smallest dwarf does not know, &nbsp;which one of these two options is in the place, &nbsp;so he can not make his suggestion.

Thus, &nbsp;if the smallest dwarf gave a gift of &nbsp;8&nbsp; flowers, &nbsp;he can make his suggestion; &nbsp;this suggestion is right, &nbsp;and this is the only 
case when the smallest dwarf can make such a suggestion.

Based in this, &nbsp;we can conclude that the smallest dwarf gave a gift of &nbsp;8&nbsp; flowers, &nbsp;and the rest of six dwarfs 
gave a gift of &nbsp;7&nbsp; flowers each.

Answer

The smallest dwarf gave a gift of &nbsp;8&nbsp; flowers.
The rest of six dwarfs gave a gift of &nbsp;7&nbsp; flowers each.

<A HREF = http://www.algebra.com/algebra/homework/word/misc/Some-logic-problems.lesson>Back to the original lesson &nbsp;Some logic problems</A>&nbsp; (the topic &nbsp;Miscellaneous word problems&nbsp; of the section &nbsp;Word problems&nbsp; in this site).