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Placing Coins in a 5x5 Matrix


Date: 04/25/2000 at 15:15:19
From: Craig George
Subject: 5 Column/5 Row Grid using coins to equal varying amounts

We have a grid consisting of 5 rows and 5 columns (25 spaces.) Each 
row and column has to equal a specific dollar amount by strategically 
placing 5 half-dollars, 5 quarters, 5 dimes, 5 nickels and 5 pennies 
in the grid. Only one coin can go in each space. On the righthand side 
row 1 = $.90, row 2 = $.53, row 3 = $2.05, row 4 = $.62, and row 5 = 
$.45. At the bottom under the columns, going from left to right, 
column one = $.72, column 2 = $1.60, column 3 = $.70, column 4 = $.86, 
and column 5 = $.67.

We have tried in vain for hours and had assistance from a relative who 
is a student teacher, with no success. Can you help?

Many thanks!


Date: 04/25/2000 at 17:45:00
From: Doctor TWE
Subject: Re: 5 Column/5 Row Grid using coins to equal varying amounts


Hi Craig - thanks for writing to Dr. Math.

This is a nice puzzle (and it is solvable!). Here's how I'd go about 
solving it.

First, I'd make some charts like these:

                                        P   N   D   Q   H
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .90     |   |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .53     |   |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   | 2.05     |   |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .62     |   |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .45     |   |   |   |   |   | 
       +---+---+---+---+---+          +===+===+===+===+===+ 
        .72 1.6 .70 .86 .67           |   |   |   |   |   | Tl.
                                      +---+---+---+---+---+
                             Tl.
       +---+---+---+---+---++---+
     P |   |   |   |   |   ||   |
       +---+---+---+---+---++---+
     N |   |   |   |   |   ||   |
       +---+---+---+---+---++---+
     D |   |   |   |   |   ||   |
       +---+---+---+---+---++---+
     Q |   |   |   |   |   ||   |
       +---+---+---+---+---++---+
     H |   |   |   |   |   ||   |
       +---+---+---+---+---++---+

The first chart is for the final placement of the coins. The chart to 
its right is for recording the number of each coin type in each row, 
and the chart below them is for recording the number of each coin type 
in each column.

Next, I'd figure out which coins have to be in each row. I'd start 
with the pennies (because it's easy to see where they have to go), 
then determine the remaining coins in those rows. For example, we know 
that the second row has to have 3 pennies (that's the only way to get 
the $.03.) Now we have to get $.50 in two coins ($.53 - $.03 = $.50 in 
5-3 = 2 coins.) The only way to do that is with 2 quarters. So we know 
the second row has 3 pennies and 2 quarters. With the other row that 
has pennies, there are two possibilities - we'll have to get more 
information. $.60 can be made from 3 coins either by 2 quarters and 1 
dime, or by 1 half-dollar and 2 nickels. Record it like this:

                                        P   N   D   Q   H
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .90     | - |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .53     | 3 | - | - | 2 | - | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   | 2.05     | - |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .62     | 2 |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .45     | - |   |   |   |   | 
       +---+---+---+---+---+          +===+===+===+===+===+ 
        .72 1.6 .70 .86 .67           | 5 |   |   | 2 |   | Tl.
                                      +---+---+---+---+---+

This means we still have to place 5 nickels, 5 dimes, 3 quarters and 
5 half-dollars. Are there any other rows we can figure out? (Hint: 
Where do most of the half-dollars have to go?) Write down the coin 
combination for that row and deduct it from the coins we still need to 
place in a row.

Now do the same thing for the columns. Start with columns that need 
pennies, then columns that don't use pennies. For example, column 5 
needs 2 pennies, and the only way to make $.65 in 3 coins is with 1 
half-dollar, 1 dime and 1 nickel. Record these as well, like this:

                                        P   N   D   Q   H
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .90     | - |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .53     | 3 | - | - | 2 | - | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   | 2.05     | - |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .62     | 2 |   |   |   |   | 
       +---+---+---+---+---+          +---+---+---+---+---+ 
       |   |   |   |   |   |  .45     | - |   |   |   |   | 
       +---+---+---+---+---+          +===+===+===+===+===+ 
        .72 1.6 .70 .86 .67           | 5 |   |   | 2 |   | Tl.
                                      +---+---+---+---+---+
                              Tl.
       +---+---+---+---+---++---+
     P | 2 | - | - | 1 | 2 || 5 |
       +---+---+---+---+---++---+
     N |   |   |   |   | 1 || 1 |
       +---+---+---+---+---++---+
     D |   |   |   |   | 1 || 1 |
       +---+---+---+---+---++---+
     Q |   |   |   |   | - || - |
       +---+---+---+---+---++---+
     H |   |   |   |   | 1 || 1 |
       +---+---+---+---+---++---+

Finally, we have to match the rows and columns up. We know that 4 of 
the 5 boxes in row 3 have to have half-dollars. Is there a column that 
does not have a half-dollar in it? If so, we know that that's where 
the nickel in row 3 goes, and the 4 half-dollars go in the remaining 
positions. By process of elimination, you should be able to place most 
or all of the coins. If you're clever, you won't have to resort to 
trial-and-error.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   
    
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