Page 106 - NUMINO Challenge_C2

Basic Concepts Frobenius Coin

Make a chart to find the amounts that can be made using a 3-cent coin and
a 5-cent coin.

Make a chart with three columns, the lowest number, and fill in the chart
with consecutive numbers.

Mark the amounts that can be made with .

Amounts that can be Amounts that can be Amounts that can be
made using two 5-cent made using two 1-cent made using 3-cent coins
coins and 3-cent coins coins and 3-cent coins
3
1 2 6
4 5 9
7 8 12
10 11 15
13 14 18
16 17

When the only common factor of the given amounts is 1, the greatest
amount that cannot be made is the product of the two amounts less the
sum of the two amounts. The greatest amount that cannot be made in the
chart above is 3 5 (3 5) 7 cents.

Example There are two types of stamps: 4-cent stamps and 5-cent stamps.
Find the greatest amount of postage that cannot be paid using the
given stamps.

Class Notes

A chart with columns starting from 1 is shown on the right. 1 2 3 4

Use the following method to find the amounts and mark the 5678
amount with on the chart. 9 10 11 12
a. All the amounts below 4 can be paid only using 4-cent 13 14 15 16
17 18 19 20
stamps. 21 22 23 24
b. All the amounts below 5 can be paid using 4-cent stamps 25 26 27 28

and one 5-cent stamp.
c. All the amounts below 10 can be paid using 4-cent stamps

and 5-cent stamps.

d. All the amounts below 15 can be paid using 4-cent stamps and 5-cent stamps.

Therefore, the greatest amount that cannot be paid using 4-cent stamps and 5-cent stamps
is cents.

103Number of Outcomes