### Brain Teasers

# Exact Postage With Three Stamps

In each of the problems below, you need to determine what stamp denominations you need to provide postage from 1 cent up to the given amount, using no more than three stamps per letter.

For example: Which two different stamps are needed to provide postage from 1 to 7 cents using no more than three stamps at a time?

Answer: you need some 1-cent and some 3-cent stamps:

1 = 1; 1 + 1 = 2; 3 = 3; 1 + 3 = 4;

1 + 1 + 3 = 5; 3 + 3 = 6; 1 + 3 + 3 = 7

(note that you can't make 8 cents without using four stamps)

Remember, for each problem below you can only use one, two, or three stamps per letter, even though you have several different types of stamps to choose from and an unlimited number of each.

First problem: With three different denominations you should be able to reach a postage of 15 cents without an interruption. What different stamps do you need?

Second problem: With four different denominations you should be able to reach a postage of 24 cents without an interruption. What stamps do you need?

Third problem: With five different denominations you should be able to reach a postage of 36 cents without an interruption. Name the stamps.

Fourth problem: With six different denominations you should be able to reach a postage of 52 cents without an interruption. (There are two sets of stamps)

Last problem (very hard and time consuming!): With seven different denominations you should be able to reach a postage of 70 cents without an interruption.

For example: Which two different stamps are needed to provide postage from 1 to 7 cents using no more than three stamps at a time?

Answer: you need some 1-cent and some 3-cent stamps:

1 = 1; 1 + 1 = 2; 3 = 3; 1 + 3 = 4;

1 + 1 + 3 = 5; 3 + 3 = 6; 1 + 3 + 3 = 7

(note that you can't make 8 cents without using four stamps)

Remember, for each problem below you can only use one, two, or three stamps per letter, even though you have several different types of stamps to choose from and an unlimited number of each.

First problem: With three different denominations you should be able to reach a postage of 15 cents without an interruption. What different stamps do you need?

Second problem: With four different denominations you should be able to reach a postage of 24 cents without an interruption. What stamps do you need?

Third problem: With five different denominations you should be able to reach a postage of 36 cents without an interruption. Name the stamps.

Fourth problem: With six different denominations you should be able to reach a postage of 52 cents without an interruption. (There are two sets of stamps)

Last problem (very hard and time consuming!): With seven different denominations you should be able to reach a postage of 70 cents without an interruption.

### Answer

First: Three stamps: 1 - 4 - 5 go up to 15Second: Four stamps: 1 - 4 - 7 - 8 up to 24

Third: Five stamps: 1 - 4 - 6 - 14 - 15 up to 36

Fourth: Six stamps: 1 - 3 - 7 - 9 - 19 - 24 (and the other set is: 1 - 4 - 6 - 14 - 17 - 29) up to 52

Last problem: Seven stamps: 1 - 4 - 5 - 15 - 18 - 27 - 34 up to 70.

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