# For Four 4's ("Student Calculator Math Book", 1980)

An exercise from the Student Calculator Maths Book, published by Texas Instruments in 1980:

Can you build all the whole numbers from 1 to 100 using only four 4's? Use only the $+$, $-$, $\times$, $\div$, $($, $)$, $.$, $x!$, $x^2$, $=$ and 4 keys on your calculator. All the whole numbers up to 119 have been built with just four 4's - how many can you find?
$1=\frac{44}{44}$
$2=\frac{4}{4}+\frac{4}{4}$
$3=(4+4+4)/4$
$4=(4!-4-4)/4$
$5=\frac{4\times 4 + 4}{4}$
$6=4+\frac{4+4}{4}$
$7=4+4-\frac{4}{4}$
$8=4-4+4+4$

Note the rule that you have to use all four of the 4's for each number (which makes it more challenging).

Additional note: David Wheeler has compiled a catalogue of 'Four fours' solutions for integers up to 40,000 (it depends on what you will allow as arithmetical operations).

Number Maths expression Calculator key presses
9 $9=4+4+\frac{4}{4}$
4 + 4 + 4 / 4
10 $10=4-4+\frac{4}{.4}$
4 - 4 + 4 / . 4
11 $11=4^2 - 4 - \frac{4}{4}$
4 x^2 - 4 - 4 / 4
12 $12=4^2 - 4 + 4 -4$
4 x^2 - 4 + 4 - 4
13 $13=4^2 - 4 + \frac{4}{4}$
4 x^2 - 4 + 4 / 4
14 $14=4^2-\frac{4+4}{4}$
4 x^2 - ( 4 + 4 ) / 4
15 $15=4\times 4 - \frac{4}{4}$
4 x 4 - 4 / 4

16 $16=\frac{4}{.4}+\frac{4!}{4}$ or $4\times 4 + 4 - 4$
4 / . 4 + 4 ! / 4
17 $17=4\times 4 + \frac{4}{4}$
4 x 4 + 4 / 4
18 $18=\frac{4}{.4}+4+4$
4 / . 4 + 4 + 4
19 $19=4! - 4 - \frac{4}{4}$
4 ! - 4 - 4 / 4
20 $20=4! - 4 + 4 - 4$
4 ! - 4 + 4 - 4

Number Maths expression Calculator key presses
21 $21=4! - 4 + \frac{4}{4}$
4 ! - 4 - 4 / 4
22 $22=4! - \frac{4+4}{4}$
4 ! - ( 4 + 4 ) / 4
23

24 $24=4! + \frac{4-4}{4}$
4 ! + ( 4 - 4 ) / 4
25


26 $26=4! + \frac{4+4}{4}$
4 ! + ( 4 + 4 ) / 4
27 $27=4! + 4 - \frac{4}{4}$
4 ! + 4 - 4 / 4
28 $28=4! + 4 + 4 - 4$
4 ! + 4 + 4 - 4
29 $27=4! + 4 + \frac{4}{4}$
4 ! + 4 + 4 / 4
30 $30= \frac{4+4+4}{.4}$
( 4 + 4 + 4 ) / . 4

Number Maths expression Calculator key presses
31 $31=4^2 + 4^2 - \frac{4}{4}$
4 x^2 + 4 x^2 - 4 / 4
32 $32=4^2 + 4^2 + 4 - 4$
4 x^2 + 4 x^2 + 4 - 4
33 $33=4^2 + 4^2 + \frac{4}{4}$
4 x^2 + 4 x^2 - 4 / 4
34 $34=44 - \frac{4}{.4}$
4 4 - 4 / . 4
35 $35 = 4! + \frac{4.4}{.4}$
4 ! + 4 . 4 / . 4
36 $36 = (4+\frac{4+4}{4})^2$
(4 + ( 4 + 4 ) / 4 ) x^2

(Work in progress...)