Four fours

Four fours is the name of a number puzzle . The aim is to represent as many numbers as possible by combining four fours with the help of arithmetic symbols. For example, the number 1 can be obtained as . ${\ displaystyle (4 + 4) / (4 + 4)}$

The puzzle exists in different variants, in particular the difficulty is varied by the selection of the allowed arithmetic symbols. Sometimes the task is also set with other digits, or more or fewer than four digits can be used.

history

A forerunner of the puzzle was published in a school book by Thomas Dilworth as early as 1743 . There he set the task of combining four threes so that the result is 34. The solution is . ${\ displaystyle 33 + 3/3}$

The puzzle was first documented in writing in 1881 in the shape of four fours. In Knowledge magazine , an unknown author - possibly the editor Richard Anthony Proctor - published the following task under the pseudonym Cupidus Scientiae : “It may be as new to some readers as it was to me when I was recently shown that all numbers up to and including twenty (and many higher), except only the nineteen, can be expressed in four fours. Any characters that require numbers can be used, with the exception of those for elevation to the second or third power. "

The puzzle quickly became popular, and variants also appeared. WW Rouse Ball systematically investigated which numbers can be represented with four identical digits. With the help of sub-faculties , among other things , he reached all numbers up to 877 with four fours, while he could not represent as many numbers with four other digits.

Nowadays the puzzle is often used as a loosening up in math lessons. It promotes arithmetic practice, trains creative and at the same time systematic searching and also introduces evidence of existence and non-existence. It is suitable for pupils of all school types from grade 5.

Arithmetic symbol

Which numbers can be represented with four fours depends on which arithmetic symbols are allowed. The permitted arithmetic symbols and other notations are usually selected from the following:

Basic arithmetic operations , brackets
Decimal notation (44), decimal point (4,4), period ( ) ${\ displaystyle 4 {,} {\ overline {4}}}$ ${\ displaystyle 4 {,} {\ overline {4}}}$
- In the English-speaking world, a decimal point is used instead, and a leading 0 can be omitted so that .4 is also possible.
Potencies ( ) ${\ displaystyle 4 ^ {4}}$
Faculty ( ) ${\ displaystyle 4!}$
Square root ( ) ${\ displaystyle {\ sqrt {4}}}$

Other functions can also be used. Even if the end result should always be an integer, any values can appear as intermediate results.

Examples

The numbers from 0 to 9 can only be represented as follows using the basic arithmetic operations:

${\ displaystyle 0 = 4 + 4-4-4}$
${\ displaystyle 1 = (4 + 4) / (4 + 4)}$
${\ displaystyle 2 = 4/4 + 4/4}$
${\ displaystyle 3 = (4 + 4 + 4) / 4}$
${\ displaystyle 4 = 4 + (4-4) / 4}$
${\ displaystyle 5 = (4 \ cdot 4 + 4) / 4}$
${\ displaystyle 6 = 4 + (4 + 4) / 4}$
${\ displaystyle 7 = 4 + 4-4 / 4}$
${\ displaystyle 8 = 4 + 4 + 4-4}$
${\ displaystyle 9 = 4 + 4 + 4/4}$

General solution

Paul Dirac found a general solution for the analog puzzle "Four Twos", provided the logarithm function is allowed:

{\ displaystyle n = - \ log _ {2} \ log _ {2} {\ sqrt [{2}] {\ sqrt {\ dots {\ sqrt {2}}}}}}

The number of nested roots is the result of the expression: The repeated extraction of the roots results in the number , the first logarithm reduces this to the exponent , the second to . If you replace every 2 with , you get a direct solution for “four fours”. ${\ displaystyle 2 ^ {2 ^ {- n}}}$ ${\ displaystyle 2 ^ {- n}}$ ${\ displaystyle -n}$ ${\ displaystyle {\ sqrt {4}}}$

Another general solution uses trigonometric functions : From the identity it results , through repeated application one obtains . This allows the number (from 5) to be represented with a 4 and repetitions of secans and arctangents . If exactly four fours are to be used, the remaining three fours can be used up, for example by adding . ${\ displaystyle \ tan ^ {2} (x) + 1 = \ sec ^ {2} (x)}$ ${\ displaystyle \ sec (\ arctan (x)) = {\ sqrt {x ^ {2} +1}}}$ ${\ displaystyle \ sec (\ arctan (\ ldots \ sec (\ arctan (x)) \ ldots)) = {\ sqrt {x ^ {2} + k}}}$ ${\ displaystyle n}$ ${\ displaystyle n ^ {2} -16}$ ${\ displaystyle 4 \ cdot (4-4)}$

Web links

Paul Bourke: 4444 problem (English)
David A. Wheeler: The Definitive Four Fours Answer Key (English)

Individual evidence

^ Thomas Dilworth: The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical . E. Duyckinck, 1810 ( limited preview in Google book search). P. 184 (A Short Collection of Pleasant and Diverting Questions), No. 9: “Says Jack to his brother Harry, I can place four threes in such a Manner that they shall just make 34; can you do so too? "

↑ Heinrich Hemme: Kopfnuss. 101 math puzzles from four millennia and five continents . 2nd Edition. Beck, Munich 2012, ISBN 3-406-63704-3 , p. 141 .

↑ Cupidus Scientiae: Four fours relation singular numerical . In: Richard Anthony Proctor (Ed.): Knowledge. An Illustrated Magazin of Science, Plainly Worded - Exactly Described . December 30, 1881, p. 184 . quoted from: Alex Bellos: What does Pythagoras have to do with garlands? The 125 best puzzles from 2000 years of mathematics . Piper, Munich, ISBN 978-3-492-06094-3 , pp. 196 (Original title: Can You Solve My Problems? Ingenious, Perplexing, and Totally Satisfying Math and Logic Puzzles . Translated by Bernhard Kleinschmidt).

↑ Alex Bellos: What does Pythagoras have to do with garlands? The 125 best puzzles from 2000 years of mathematics . Piper, Munich, ISBN 978-3-492-06094-3 , pp. 195-201 .

^ WW Rouse Ball: Mathematical recreations and essays . Macmillan, London 1917, pp. 14 ( archive.org ).

↑ Four fours, four twos, and other number term mental exercises. SINUS , accessed July 1, 2019 .

↑ Christina Drüke-Noe, Dominik Leiß: Standard mathematics from the base to the top - basic education-oriented tasks for mathematics lessons . In: Hessian State Institute for Education (Ed.): Materials for teaching secondary . Frankfurt am Main 2004, ISBN 3-88327-516-6 ( hessen.de [PDF]).

↑ Hendrik Casimir : Haphazard Reality: Half a Century of Science . Amsterdam University Press, 2010, ISBN 978-90-8964-200-4 , pp. 75 ( limited preview in Google Book search).

↑ Graham Farmelo: The Strangest Man: The Hidden Life of the Quantum Genius Paul Dirac . Springer-Verlag, 2018, ISBN 978-3-662-56579-7 , pp. 181 ( limited preview in Google Book search).

↑ Jim Millar: Four Fours problem: Any number with One Four. 2006, accessed July 1, 2019 .

[1] Thomas Dilworth: The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical . E. Duyckinck, 1810 ( limited preview in Google book search). P. 184 (A Short Collection of Pleasant and Diverting Questions), No. 9: “Says Jack to his brother Harry, I can place four threes in such a Manner that they shall just make 34; can you do so too? "

[2] Heinrich Hemme: Kopfnuss. 101 math puzzles from four millennia and five continents . 2nd Edition. Beck, Munich 2012, ISBN 3-406-63704-3 , p. 141 .

[3] Cupidus Scientiae: Four fours relation singular numerical . In: Richard Anthony Proctor (Ed.): Knowledge. An Illustrated Magazin of Science, Plainly Worded - Exactly Described . December 30, 1881, p. 184 . quoted from: Alex Bellos: What does Pythagoras have to do with garlands? The 125 best puzzles from 2000 years of mathematics . Piper, Munich, ISBN 978-3-492-06094-3 , pp. 196 (Original title: Can You Solve My Problems? Ingenious, Perplexing, and Totally Satisfying Math and Logic Puzzles . Translated by Bernhard Kleinschmidt).

[4] Alex Bellos: What does Pythagoras have to do with garlands? The 125 best puzzles from 2000 years of mathematics . Piper, Munich, ISBN 978-3-492-06094-3 , pp. 195-201 .

[5] WW Rouse Ball: Mathematical recreations and essays . Macmillan, London 1917, pp. 14 ( archive.org ).

[6] Four fours, four twos, and other number term mental exercises. SINUS , accessed July 1, 2019 .

[7] Christina Drüke-Noe, Dominik Leiß: Standard mathematics from the base to the top - basic education-oriented tasks for mathematics lessons . In: Hessian State Institute for Education (Ed.): Materials for teaching secondary . Frankfurt am Main 2004, ISBN 3-88327-516-6 ( hessen.de [PDF]).

[8] Hendrik Casimir : Haphazard Reality: Half a Century of Science . Amsterdam University Press, 2010, ISBN 978-90-8964-200-4 , pp. 75 ( limited preview in Google Book search).