can not understand this Puzzle :(

[zenith20]

Hi there,
i have recently read the book "puzzles to puzzle you" from shakuntala devi, but i didnt understand some of them. for instance the following question:

"A wholesale merchant came to me one day and posed this problem.Everyday in his business he had to weigh amounts fromm one pound to hundred and twenty-one pounds, to the nearest pound.To do this, what is the minimum number of weighs he needs and how heavy should each weight be?"

actually i didnt understand the question itself, esp the part "to the nearest pound"!

and the solution is:

"?The minimum number of weights required is five and these should weigh 1, 3, 9, 27 and 81 pounds."

Could you please explain me both the question and the solution.

Thank you in advance,
Zenith

 

[Deleted member 4993]

Hi there,
i have recently read the book "puzzles to puzzle you" from shakuntala devi, but i didnt understand some of them. for instance the following question:

"A wholesale merchant came to me one day and posed this problem.Everyday in his business he had to weigh amounts fromm one pound to hundred and twenty-one pounds, to the nearest pound.To do this, what is the minimum number of weighs he needs and how heavy should each weight be?"

actually i didnt understand the question itself, esp the part "to the nearest pound"!

and the solution is:

"?The minimum number of weights required is five and these should weigh 1, 3, 9, 27 and 81 pounds."

Could you please explain me both the question and the solution.

Thank you in advance,
Zenith

First we have to assume the merchant is going to use a two-pan balance for weighing.

Understanding the answer is simple.

To the nearest pound means - his weights will be all integers. He is not going to weigh 1/2 pound or 3.14159 pounds etc.

Then

suppose he needs weigh 2 pounds - he'll put 3 pound weight on one pan and 1 pound weight on the other pan.

suppose he needs weigh 100 pounds - he'll put (81+27+1) pound weight on one pan and 9 pound weight on the other pan.

You can show that any integer - from 1 to 121 can be made by using those 4 numbers (only once) and using addition or subtraction.

As for myself - I do not know the steps to the solution. I knew the solution (somebody told me) in grade school and it kind'a stuck with me.

 

[zenith20]

Dear Subhotosh Khan,
thanks for your prompt reply. now that's much more understood

have a great time,
Zenith

 

[BigGlenntheHeavy]

"As for myself - I do not know the steps to the solution." quote Subhotosh Khan.

\(\displaystyle Note: \ neither \ do \ I, \ but \ I \ noticed \ that \ 3^0+3^1+3^2+3^3+3^4 \ = \ 121\)

\(\displaystyle A \ series? \ any \ takers \ from \ here?\)

 

[soroban]

Hello, zenith20!

Subhotosh is absolutely correct!

\(\displaystyle \text{This balance-scale problem can be solved in a base-3 (ternary) system.}\)

\(\displaystyle \text{The required weights are powers-of-3: }\;1,\:3,\:9,\:27,\:81,\:\hdots\)

\(\displaystyle \text{To weigh some commodity (in integer pounds) , we place the commodity in one pan of the scale}\)
. . \(\displaystyle \text{and weights in one or both pans so that the pans balance.}\)


\(\displaystyle \text{To weigh 93 pounds, convert 93 to base-3: }\;93 \;=\;10111_3\)

\(\displaystyle \text{This ternary number represents: }\;\begin{array}{ccccc} \\ 1 & 0 & 1 & 1 & 0 \\ [-2mm] - & - & - & - & - \\ [-1mm] ^{81} & ^{27} & ^9 & ^3 & ^1 \end{array}\)

\(\displaystyle \text{This means: place the 81, 9, and 3 in the left pan . . . and the commodity on the right.}\)

. . \(\displaystyle \begin{array}{ccc} \boxed{81}\,\boxed{9}\,\boxed{3} & & \boxed{C} \\ \\[-3mm] \hline \\ [-4mm] & \wedge \end{array}\)



\(\displaystyle \text{But what if a "2" appears in the ternary number?}\)
. . \(\displaystyle \text{There is a special trick for this situation.}\)

\(\displaystyle \text{Example: }\;100 \:=\:10201_3\)

\(\displaystyle \text{This represents: }\;\begin{array}{ccccc} \\ 1&0&2&0&1 \\ [-2mm] - & - & - & - & - \\ [-1mm] ^{81} & ^{27} & ^9 & ^3 & ^1 \end{array}\)

\(\displaystyle \text{Replace the "2" with "-1", and add 1 to the position to the immediately left:}\)

. . . . . . . . . . . . \(\displaystyle \begin{array}{ccccc} 1 & 1 & \text{-}1 & 0 & 1 \\ [-2mm]- & - & - & - & - \\ [-1mm] ^{81} & ^{27} & ^9 & ^3 & ^1 \end{array}\)


\(\displaystyle \text{This is interpreted like this: }\;\begin{array}{c}\text{Place the 81, 27, and 1 in the left pan.} \\ \text{Place 9 and the commodity in the right pan.} \end{array}\)

. . \(\displaystyle \begin{array}{ccc} \boxed{81}\;\boxed{27}\;\boxed{1} && \boxed{9}\;\boxed{C} \\ \\[-3mm] \hline \\ [-4mm] & \wedge \end{array}\)

 

[Deleted member 4993]

Hello, zenith20!

Subhotosh is absolutely correct!

I got to correct that - I think you meant BigGlenn

 

[zenith20]

i need to say THANKS A LOT to EVERYONE

Dear Soroban, your detailed answer is really appreciated.

MANY MANY THANKS

Zenith