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Chapter 1: Problem 60

American coins are all marked with the year in which they were made. How many coins do you need to have in your hand to guarantee that on two (at least) of them, the date has the same last digit? (When we say "to guarantee that on two (at least) of them,..." we mean that you can find two with the same last digit. You might be able to find three with that last digit, or you might be able to find one pair with the last digit 1 and one pair with the last digit \(9,\) or any combination of equal last digits, as long as there is at least one pair with the same last digit.)

Short Answer

Expert verified
At least 11 coins are needed.

Step by step solution

01

Understand the problem

Determine how many coins are needed to guarantee that at least two coins have dates ending with the same last digit.
02

Identify the range of last digits

American coins can have dates ending in any of the digits from 0 to 9. Therefore, there are 10 possible last digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
03

Use the pigeonhole principle

According to the pigeonhole principle, if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. Here, the 'pigeons' are the coins, and the 'pigeonholes' are the possible last digits.
04

Calculate the minimum number of coins needed

To ensure at least two coins share the same last digit, we must have more coins than there are unique last digits. Since there are 10 unique last digits, we need at least 11 coins. With 11 coins, we are guaranteed that at least two coins will share the same last digit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of objects. In this exercise, combinatorics helps us understand how to distribute coins based on their last digits. The pigeonhole principle, a fundamental concept in combinatorics, states that if you have more items than containers, at least one container must hold more than one item. For our problem, the 'items' are the coins and the 'containers' are the possible last digits (0 through 9). With 10 unique last digits, having at least 11 coins means one digit will surely repeat.
Number Theory
Number theory explores the properties and relationships of numbers, especially integers. A key number theory concept in this problem is the 'last digit' of a number, which provides a simplified way to classify and compare numbers. By focusing on the last digit of the coin's year, we reduce the problem's complexity. Each coin's year can end in one of 10 digits (0-9), forming a finite and manageable set. Thus, understanding modular arithmetic, or congruence, helps in dealing with repetitive cycles like last digits.
Problem Solving
Problem solving in mathematics involves understanding the problem, devising a plan, carrying out the plan, and reviewing/reflecting on the solution. Let's break down the steps for this exercise:

1. **Understand the Problem:** We need to guarantee at least two coins have the same last digit.

2. **Devise a Plan:** Use the pigeonhole principle to determine the minimum number of coins needed.

3. **Carry Out the Plan:** Identify that there are 10 possible last digits (0-9). Calculate using the pigeonhole principle: with 11 coins, at least two must share the same last digit.

4. **Review the Solution:** Verify by ensuring all digits 0-9 can be unique with up to 10 coins. Adding the 11th coin forces a repeat of one digit.

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Most popular questions from this chapter

There is a generalized pigeonhole principle which says that if we partition a set with more than \(k n\) elements into \(n\) blocks, then at least one block has at least \(k+1\) elements. Prove the generalized pigeonhole principle. (h)

One of the schools sending its team to the tournament has to send its players from some distance, and so it is making sandwiches for team members to eat along the way. There are three choices for the kind of bread and five choices for the kind of filling. How many different kinds of sandwiches are available? (b)

In how many ways can we pass out \(k\) (identical) ping-pong balls to \(n\) children if each child may get at most one? (h)

What is \(\left(\begin{array}{c}n \\\ 0\end{array}\right)-\left(\begin{array}{c}n \\\ 1\end{array}\right)+\left(\begin{array}{c}n \\ 2\end{array}\right)-\cdots \pm\left(\begin{array}{c}n \\ n\end{array}\right)\) if \(n\) is an integer bigger than \(0 ?(\mathrm{~h})\)

Another name for a list, in a specific order, of \(k\) distinct things chosen from a set \(S\) is a \(k\) -element permutation of \(S .\) We can also think of a \(k\) -element permutation of \(S\) as a one-to-one function (or, in other words, injection) from \([k]=\\{1,2, \ldots, k\\}\) to \(S .\) How many \(k\) -element permutations does an \(n\) -element set have? (For this problem it is natural to assume \(k \leq n\). However the question makes sense even if \(k>n\). What is the number of \(k\) -element permutations of an \(n\) -element set if \(k>n ?\)
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