Four fours
Tonight's idea
The puzzle: use exactly four 4s, with \(+\), \(-\), \(\times\), \(\div\) and brackets, to make each whole number from 0 to 20.
Two 4s make only four values: \(4 - 4 = 0\), \(4 \div 4 = 1\), \(4 + 4 = 8\) and \(4 \times 4 = 16\). These are building blocks. Join two blocks and you have used four 4s.
Order matters. Multiply and divide before you add and subtract, unless brackets say otherwise (grade 6). So \(4 + 4 \times 4 - 4\) is \(4 + 16 - 4 = 16\), but \((4 + 4) \times (4 - 4)\) is \(8 \times 0 = 0\). Brackets are a tool.
A computer search of every arrangement shows this reaches 15 of the 21 targets. The other six are 10, 11, 13, 14, 18 and 19. For those, two more tools are allowed.
\(\sqrt4 = 2\), the square root of 4, because \(2 \times 2 = 4\). And \(4! = 4 \times 3 \times 2 \times 1 = 24\), read "4 factorial". Each one still uses up one 4.
Watch one: making 7, then 13
7 is one less than 8. Take the blocks \(4 + 4 = 8\) and \(4 \div 4 = 1\):
\[4 + 4 - 4 \div 4 = 8 - 1 = 7.\]No brackets are needed: the division happens first.
13 is on the hard list. Now \(4! \div \sqrt4 = 24 \div 2 = 12\), and \(4 \div 4 = 1\):
\[4! \div \sqrt4 + 4 \div 4 = 12 + 1 = 13.\]Always count the 4s: exactly four.
Your turn
- Work out each one. Take care with the order. (a) \(4 + 4 \div 4 + 4\) (b) \(4 \times 4 \div 4 \div 4\) (c) \(4 \times 4 - 4 - 4\) (d) \((4 + 4 + 4) \div 4\)
- Use four 4s, with only \(+\), \(-\), \(\times\), \(\div\) and brackets, to make (a) 0 (b) 2 (c) 6 (d) 12.
- With no brackets, \(4 \times 4 + 4 \div 4\) is 17. Keep the 4s and signs in the same order, and add brackets to make (a) 20 (b) 5 (c) 8.
- Now \(\sqrt4\) and \(4!\) are allowed. Make (a) 10 (b) 14 (c) 18 (d) 19. For 19, think of a number close to it that you can make from one 4.
Stretch: three 4s
Use exactly three 4s, with only \(+\), \(-\), \(\times\), \(\div\) and brackets. Which of the numbers 0 to 6 can you make? Two of them are impossible. Find the two, and explain how you can be sure without trying for ever.
Answers
- (a) \(4 + 1 + 4 = 9\). (b) \(16 \div 4 \div 4 = 1\). (c) \(16 - 4 - 4 = 8\). (d) \(12 \div 4 = 3\).
- See the table. Other answers are possible: check yours by working it out.
- (a) \(4 \times (4 + 4 \div 4) = 4 \times 5 = 20\). (b) \((4 \times 4 + 4) \div 4 = 20 \div 4 = 5\). (c) \(4 \times (4 + 4) \div 4 = 32 \div 4 = 8\).
- See the table. For 19, start from \(4! = 24\) and take away 5.
One way to make each number. Every line has been checked by computer.
| 0 | \(4 + 4 - 4 - 4\) | 7 | \(4 + 4 - 4 \div 4\) | 14 | \(4 + 4 + 4 + \sqrt4\) |
| 1 | \((4 + 4) \div (4 + 4)\) | 8 | \(4 + 4 + 4 - 4\) | 15 | \(4 \times 4 - 4 \div 4\) |
| 2 | \(4 \div 4 + 4 \div 4\) | 9 | \(4 + 4 + 4 \div 4\) | 16 | \(4 + 4 + 4 + 4\) |
| 3 | \((4 + 4 + 4) \div 4\) | 10 | \(4 + 4 + 4 - \sqrt4\) | 17 | \(4 \times 4 + 4 \div 4\) |
| 4 | \(4 + 4 \times (4 - 4)\) | 11 | \(4 + (4! + 4) \div 4\) | 18 | \(4 \times 4 + 4 - \sqrt4\) |
| 5 | \((4 \times 4 + 4) \div 4\) | 12 | \(4 \times (4 - 4 \div 4)\) | 19 | \(4! - 4 - 4 \div 4\) |
| 6 | \(4 + (4 + 4) \div 4\) | 13 | \(4! \div \sqrt4 + 4 \div 4\) | 20 | \(4 \times (4 + 4 \div 4)\) |
Stretch. \((4 - 4) \times 4 = 0\), \((4 + 4) \div 4 = 2\), \(4 - 4 \div 4 = 3\), \(4 + 4 - 4 = 4\), \(4 + 4 \div 4 = 5\). Impossible: 1 and 6. To be sure, note that three 4s are always one block of two 4s (0, 1, 8 or 16) joined to a single 4. That is only four blocks, four signs and two orders, so you can list every result. The whole numbers on the list are −12, −4, −3, 0, 2, 3, 4, 5, 12, 20, 32 and 64.