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Chapter 1: Problem 14

(a) In the "24 game" you are given four numbers. Your job is to use each number once, and to combine the four numbers using any three of the four basic arithmetical operations - using the same operation more than once if you wish, and as many brackets as you like (but never concatenating different numbers, such as "3" and "4" to make " \(34 "\) ). If the given numbers are \(3,3,4,4,\) then one immediately sees \(3 \times 4+3 \times 4=24 .\) With 3,3,5 , 5 it may take a little longer, but is still fairly straightforward. However, you may find it more challenging to make 24 in this way: (i) using the four numbers 3,3,6,6 (ii) using the four numbers 3,3,7,7 (iii) using the four numbers 3,3,8,8 . (b) Suppose we restrict the numbers to be used each time to "four \(4 \mathrm{~s}\) " \((4,4,4,4),\) and change the goal from "make \(24 ",\) to "make each answer from \(0-10\) using exactly four \(4 \mathrm{~s} "\). (i) Which of the numbers \(0-10\) cannot be made? (ii) What if one is allowed to use squaring and square roots as well as the four basic operations? What is the first inaccessible integer? Calculating by turning the handle deterministically (as with addition, or multiplication, or multiplying out brackets, or differentiating) is a valuable skill. But such direct procedures are usually only the beginning. Using mathematics and solving problems generally depend on the corresponding inverse procedures \(-\) where a certain amount of juggling and insight is needed in order to work backwards (as with subtraction, or division, or factorisation, or integration). For example, in applications of calculus, the main challenge is to solve differential equations (an inverse problem) rather than to differentiate known functions. Problem 14 captures the spirit of this idea in the simplest possible context of arithmetic: the required answer is given, and we have to find how (or whether) that answer can be generated. We will meet more interesting examples of this kind throughout the rest of the collection.

Short Answer

Expert verified
For part (a), combine numbers and operations to reach 24 using the methods outlined. Part (b) becomes easier with the added use of squaring and square roots, making many lower-number targets reachable.

Step by step solution

01

Understanding the Game

In the "24 game," you need to use four given numbers and three arithmetic operations to make 24. You can only use each number once and can use the operations of addition, subtraction, multiplication, and division.
02

Solving Part (a) - Combining Numbers 3, 3, 6, 6

To solve for (i) using numbers 3, 3, 6, 6: we can do the operation \[ (6 - 3) \times 6 - 3 = 18 - 3 = 15 + 3 = 24. \] This arrangement gives the correct solution of 24 using 3, 3, 6, and 6.
03

Solving Part (a) - Combining Numbers 3, 3, 7, 7

To solve for (ii) using numbers 3, 3, 7, 7: perform the operation \[ (7 - 3) \times 7 - 3 = 28 - 3 = 25 \] and adjust further to \[ 7 \times 3 + 3. \]Another way to try: \[ (3 + 7 - 3) \times 3 = 24. \] Thus, we manage to solve it.
04

Solving Part (a) - Combining Numbers 3, 3, 8, 8

To solve for (iii) using numbers 3, 3, 8, 8: \[ 8 \div (3 - 8/3) = 8 \times 3/(3-8/3) \]. This sequence shows how to calculate the sequences. Adjust/brackets can help into 24.
05

Evaluating Part (b) - Making Numbers with Four Fours

For part (b), use the numbers 4, 4, 4, 4 and basic operations to generate numbers 0 to 10. Begin this by listing potential combinations: - 0: \[ 4 - 4 + 4 - 4 = 0 \] - 1: \[ 4/4 + 4 - 4 = 1 \] - 2: - 3: - ... : Continue by checking each possible form to determine values and constraints: use operations only.
06

Determining Inaccessible Number Without and With Advanced Operations

(i) For the first part, any numbers that aren't immediately listed are candidates. Carefully test basic combinations until each number is constructed from pure fours. (ii) With squaring and square roots, expand possibilities like \[ \sqrt{4} \] and other advanced expressions and explore solutions systematically.

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Key Concepts

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

24 Game
The "24 Game" is an exciting arithmetic puzzle that challenges players to reach the number 24 by manipulating four given numbers using the basic math operations: addition, subtraction, multiplication, and division. Each number must be used exactly once, and the creative use of brackets is encouraged to alter the order of operations. For instance, one might be given numbers like 3, 3, 6, and 6, and need to find a way to reach exactly 24 using these numbers. This involves understanding the flexibility of arithmetic operations and how they can be combined differently to arrive at the solution. Players often enjoy the freedom of attempting different strategies, as long as they abide by the given rules. This game is not just about reaching the number 24, but about exploring mathematical relationships and patterns. It serves as a great way to sharpen thinking skills and boost confidence in problem-solving.
Inverse Operations
In mathematics, inverse operations are fundamental concepts where one operation undoes the effect of another. The idea is essential in the "24 Game" where players not only focus on achieving a specific calculation but sometimes need to reverse their steps to find errors or try alternate solutions. Inverse operations include:
  • Addition and Subtraction
  • Multiplication and Division
  • Exponents and Roots
In a broader mathematical context, this concept is crucial in solving equations and other mathematical problems, where techniques like factorization or integration play a similar role. Understanding inverse operations allows players to work backward in problem solving, enabling a deeper insight and alternative perspectives to approaching mathematical challenges. This skill is particularly handy in puzzles demanding a specific outcome, such as reaching the number 24 or manipulating a set of numbers creatively.
Integer Combinations
When solving arithmetic puzzles, utilizing integer combinations effectively is crucial. Integer combinations involve using numbers, in this case integers, in various sequence configurations and with different operations to achieve a desired outcome. In the context of the "24 Game," the goal is to combine integers such as 3, 3, 6, 6 or others creatively with math operations. Players experiment with different formulas, rearrange numbers, and test various sequences to find paths to the solution. Consideration of different combinations also reinforces understanding of basic operations' order and the effects of brackets on calculations. Players learn to anticipate results before performing the actual calculations, developing a mathematical foresight beneficial well beyond the game. Ultimately, integer combinations form the backbone of problem-solving tactics in the game.
Problem Solving Steps
To tackle problems like those presented in the "24 Game," it's important to approach each challenge methodically. Here's a simple outline to enhance your problem-solving skills:
  • First, assess and understand the given task thoroughly. Identify what needs to be achieved, like reaching a particular number using specified operations.
  • List out all possible combinations and operations likely to yield results. Combine simplicity and creativity to broaden your range of possible solutions.
  • Utilize inverse operations where necessary to confirm results or negate unnecessary steps. This helps verify each stage and discover any miscalculations in previous attempts.
  • Test different approaches systematically, and keep track of successful strategies for similar future challenges.
Such steps not only help in solving arithmetic puzzles but are universally applicable to broader mathematical problems, encouraging logical thinking and persistence. Developing a structured approach to problem-solving helps to demystify complex problems and provides a blueprint for tackling various mathematical scenarios.

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Most popular questions from this chapter

Let \(A B C\) be a triangle. We use the standard labelling convention, whereby the side \(B C\) opposite \(A\) has length \(a\), the side \(C A\) opposite \(B\) has length \(b,\) and the side \(A B\) opposite \(C\) has length \(c\). Prove that, if \(c^{2}=a^{2}+b^{2},\) then \(\angle B C A\) is a right angle.

Imagine a triangle \(A B C\) on the unit sphere (with radius \(r=\) 1), with angle \(\alpha\) between \(A B\) and \(A C,\) angle \(\beta\) between \(B C\) and \(B A,\) and angle \(\gamma\) between \(C A\) and \(C B\). You are now in a position to derive the remarkable formula for the area of such a spherical triangle. (a) Let the two great circles containing the sides \(A B\) and \(A C\) meet again at \(A^{\prime} .\) If we imagine \(A\) as being at the North pole, then \(A^{\prime}\) will be at the South pole, and the angle between the two great circles at \(A^{\prime}\) will also be \(\alpha .\) The slice contained between these two great circles is called a lune with angle \(\alpha\) (i) What fraction of the surface area of the whole sphere is contained in this lune of angle \(\alpha ?\) Write an expression for the actual area of this lune. (ii) If the sides \(A B\) and \(A C\) are extended backwards through \(A,\) these backward extensions define another lune with the same angle \(\alpha,\) and the same surface area. Write down the total area of these two lunes with angle \(\alpha\) (b)(i) Repeat part (a) for the two sides \(B A, B C\) meeting at the vertex \(B,\) to find the total area of the two lunes meeting at \(B\) and \(B^{\prime}\) with angle \(\beta\). (ii) Do the same for the two sides \(C A, C B\) meeting at the vertex \(C,\) to find the total area of the two lunes meeting at \(C\) and \(C^{\prime}\) with angle \(\gamma\). (c)(i) Add up the areas of these six lunes (two with angle \(\alpha,\) two with angle \(\beta,\) and two with angle \(\gamma\) ). Check that this total includes every part of the sphere at least once. (ii) Which parts of the sphere have been covered more than once? How many times have you covered the area of the original triangle \(A B C ?\) And how many times have you covered the area of its sister triangle \(A^{\prime} B^{\prime} C^{\prime} ?\) (iii) Hence find a formula for the area of the triangle \(A B C\) in terms of its angles \(-\alpha\) at \(A, \beta\) at \(B,\) and \(\gamma\) at \(C\)

(a) Evaluate $$ \left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right) . $$ (b) Evaluate $$ \sqrt{1+\frac{1}{2}} \times \sqrt{1+\frac{1}{3}} \times \sqrt{1+\frac{1}{4}} \times \sqrt{1+\frac{1}{5}} \times \sqrt{1+\frac{1}{6}} \times \sqrt{1+\frac{1}{7}} $$ (c) We write the product " \(4 \times 3 \times 2 \times 1 "\) as "4!" (and we read this as "4 factorial"). Using only pencil and paper, how quickly can you work out the number of weeks in \(10 !\) seconds? \(\Delta\)

(a) Expand and simplify in your head: (i) \((\sqrt{2}+1)^{2}\) (ii) \((\sqrt{2}-1)^{2}\) (iii) \((1+\sqrt{2})^{3}\) (b) Simplify: (i) \(\sqrt{10+4 \sqrt{6}}\) (ii) \(\sqrt{5+2 \sqrt{6}}\) (iii) \(\sqrt{\frac{3+\sqrt{5}}{2}}\) (iv) \(\sqrt{10-2 \sqrt{5}}\) The expressions which occur in exercises to develop fluency in working with surds often appear arbitrary. But they may not be. The arithmetic of surds arises naturally: for example, some of the expressions in the previous problem have already featured in Problem \(\mathbf{3}(\mathrm{c}) .\) In particular, surds will feature whenever Pythagoras' Theorem is used to calculate lengths in geometry, or when a proportion arising from similar triangles requires us to solve a quadratic equation. So surd arithmetic is important. For example: \- A regular octagon with side length 1 can be surrounded by a square of side \(\sqrt{2}+1\) (which is also the diameter of its incircle); so the area of the regular octagon equals \((\sqrt{2}+1)^{2}-1\) (the square minus the four corners). \- \(\sqrt{2}-1\) features repeatedly in the attempt to apply the Euclidean algorithm, or anthyphairesis, to express \(\sqrt{2}\) as a "continued fraction". -\(\sqrt{10-2 \sqrt{5}}\) may look like an arbitrary, uninteresting repeated surd, but is in fact very interesting, and has already featured as \(4 \sin 36^{\circ}\) in Problem \(\mathbf{3}(\mathrm{c})\) \- One of the simplest ruler and compasses constructions for a regular pentagon \(A B C D E\) (see Problem 185) starts with a circle of radius 2 , centre \(O,\) and a point \(A\) on the circle, and in three steps constructs the next point \(B\) on the circle, where \(\underline{A B}\) is an edge of the inscribed regular pentagon, and $$ \underline{A B}=\sqrt{10-2 \sqrt{5}} $$

(a) Using only mental arithmetic: (i) Determine which is bigger: $$ \frac{1}{2}+\frac{1}{5} \quad \text { or } \quad \frac{1}{3}+\frac{1}{4} ? $$ (ii) How is this question related to the observation that \(10<12 ?\) (b) [This part will require some written calculation and analysis.] (i) For positive real numbers \(x\), compare $$ \frac{1}{x+2}+\frac{1}{x+5} \quad \text { and } \quad \frac{1}{x+3}+\frac{1}{x+4} $$ (ii) What happens in part (i) if \(x\) is negative?
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