Coin Distribution

 

This activity consists of two parts. One is entitled Are There Any Dimes in a Piggy Bank?, and the other is based on a classic partition problem asking for the number of ways you can make change for a dollar using pennies, nickels, dimes, and quarters.

Piggy Bank

The activity has been designed and is in working order, but we are not happy with the appearance of the page. This will be drastically changed. Applets have also been designed for more general cases (not exhibited here).

Coin Distribution

This activity arises in a natural way from the Piggy Bank activity. The basic question to be treated is:

In how many ways can one produce a given sum?

Applets have been prepared in the form of games that can be played at different levels. At level k the number of coins of each type cannot exceed k. The default setting is level 10, but the user can adjust this to a different level. In the first applet, the user is asked to create a prescribed sum with a given number of coins at the chosen level. The sum and number of coins are selected at random by the computer in such a way that a solution is always possible at the level chosen by the user. Images of the coins (pennies, nickels, dimes, quarters) can be brought to the screen by clicking, and they can be seen as stacked or spread out. The user clicks on the number of each type of coin candidate, and the computer responds with the resulting sum for that choice of coins (which may or may not be the prescribed sum). If the user makes a correct selection, the computer says so. If you click on (?) the computer will show a list of all solutions at that level.

A second applet allows the viewer to enter the desired sum and the total number of coins. Then the game is played as before, except that no solution may exist at the chosen level.

In the next applet the user selects the total number of coins and the sum, but there is no restriction on the level. The game proceeds as before, with the computer showing all possible solutions with the selected sum and number of coins.

A further applet shows that, for a particular known solution, replacement of coins can be made to find another solution with the same sum and the same number of coins. For example, one quarter and three nickels (four coins) can be replaced by four dimes, which give the same total.