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### Combining Numbers

```Date: 08/04/2002 at 15:36:00
From: Roger
Subject: A problem from my daughter's tutor!

Hi,

Combine the numbers 3, 3, 8, and 8, using only addition, subtraction,
multiplication, and division, to come up with the number 24.

We appear to have used every combination and have not come up with
the answer 24.  We just need to know how it is worked out please.

Warm regards,
Roger
```

```
Date: 08/05/2002 at 18:14:19
From: Doctor Ian
Subject: Re: A problem from my daughter's tutor!

Hi Roger,

What makes this kind of problem so difficult is that it's easy to just
start trying things, without any good way of keeping track of what
you've tried, or any way of knowing when you've tried everything.

Ultimately, all the operators (+, -, *, /) are binary, which means
they have to work on two operands, so we have to begin by choosing two
operands to work with.  There are three ways we can do that:

(3,3,8,8) -> ((3,3),8,8)
((3,8),3,8)
((8,8),3,3)

Here are the results we can get by applying one operator to these
pairs of operands:

(a,b)    a+b  a-b  b-a  a*b  a/b  b/a   Remaining

(3,3) ->   6    0    0    9    1    1   (8,8)

(3,8) ->  11   -5    5   24  3/8  8/3   (3,8)

(8,8) ->  16    0    0   64    1    1   (3,3)

Note that we can use each of the numbers in the center as a new
starting point for a smaller version of the same problem:

(3,3,8,8) -> ((3,3),8,8) => (6,8,8)
(0,8,8)
(9,8,8)
(1,8,8)

((3,8),3,8) => (11,3,8)
(-5,3,8)
(5,3,8)
(24,3,8)
(3/8,3,8)
(8/3,3,8)

((8,8),3,3) => (16,3,3)
(0,,3,3)
(64,3,3)
(1,3,3)

Now we can do the same thing (i.e., pair the remaining operands
in various ways) again:

(3,3,8,8) -> ((3,3),8,8) => (6,8,8)   -> ((6,8),8)
((8,8),6)

(0,8,8)   -> ((0,8),8)
((8,8),0)

(9,8,8)   -> ((9,8),8)
((8,8),9)

(1,8,8)   -> ((1,8),8)
((8,8),1)

((3,8),3,8) => (11,3,8)  -> ((11,3),8)
((11,8),3)
((3,8),11)

(-5,3,8)  -> ((-5,3),8)
((-5,8),3)
((3,8),-5)

(5,3,8)   -> ((5,3),8)
((5,8),3)
((3,8),5)

(24,3,8)  -> ((24,3),8)
((24,8),3)
((3,8),24)

(3/8,3,8) -> ((3/8,3),8)
((3/8,8),3)
((3,8),3/8)

(8/3,3,8) -> ((8/3,3),8)
((8/3,8),3)
((3,8),8/3)

((8,8),3,3) => (16,3,3)  -> ((16,3),3)
((3,3),16)

(0,3,3)   -> ((0,3),3)
((3,3),0)

(64,3,3)  -> ((64,3),3)
((3,3),64)

(1,3,3)   -> ((1,3),3)
((3,3),1)

For the next step, we need to make up another table for the distinct
pairs in the rightmost column:

(a,b)      a+b   a-b   b-a   a*b   a/b   b/a     Remaining

(-5,3) ->   -2    -8     8   -15  -5/3  -3/5     8
(-5,8) ->   -3   -13    13   -40  -5/8  -8/5     3
(0,3) ->    3    -3     3     0     0    ~      3
(0,8) ->    8    -8     8     0     0    ~      8
(3/8,3) -> 27/8 -21/8  21/8   9/8  3/24     8     8
(3/8,8) -> 67/8 -61/8  61/8     3  3/64  64/3     3
(1,3) ->    4    -2     2     3   1/3     3     3
(1,8) ->    9    -7     7     8   1/8     8     8
(8/3,3) -> 17/3  -1/3   1/3     8   8/9   9/8     8
(8/3,8) -> 32/3 -16/3  16/3  64/3  8/24     3     3
(3,3) ->    6     0     0     9     1     1     0, 1, 16, 64
(3,8) ->   11    -5     5    24   3/8   8/3     -5, 3/8, 8/3,
5, 11, 24
(5,3) ->    8     2    -2    15   5/3   3/5     8
(5,8) ->   13    -3     3    40   5/8   8/5     3
(6,8) ->   14    -2     2    48   6/8   8/6     8
(8,8) ->   16     0     0    64     1     1     0, 1, 6, 9
(9,8) ->   17     1    -1    72   9/8   9/8     8
(11,3) ->   14     8    -8    33  11/3  3/11     8
(11,8) ->   19     3    -3    88  11/8  8/11     3
(16,3) ->   19    13   -13    48  16/3  3/16     3
(24,3) ->   27    21   -21    72     8   1/8     8
(24,8) ->   32    16   -16   192     3   1/3     3
(64,3) ->   67    61   -61   192  64/3  3/64     3

Okay, so what can we do with this?  Well, the _only_ way we can end up
with 24 is if we can do it in one by adding, subtracting, multiplying,
or dividing something from the Remaining column in a given row with
something else in the row.

For example, let's look at the row

(a,b)      a+b   a-b   b-a   a*b   a/b   b/a     Remaining

(-5,3) ->   -2    -8     8   -15  -5/3  -3/5     8

There is no way to combine 8 with -2, -8, 8, -15, -5/3, or -3/5 to end
easy ones:

(a,b)      a+b   a-b   b-a   a*b   a/b   b/a     Remaining

(-5,3) ->   -2    -8     8   -15  -5/3  -3/5     8 (No)
(-5,8) ->   -3   -13    13   -40  -5/8  -8/5     3 (No)
(0,3) ->    3    -3     3     0     0    ~      3 (No)
(0,8) ->    8    -8     8     0     0    ~      8 (No)
(3/8,3) -> 27/8 -21/8  21/8   9/8  3/24     8     8 (No)
(3/8,8) -> 67/8 -61/8  61/8     3  3/64  64/3     3 (No)
(1,3) ->    4    -2     2     3   1/3     3     3 (No)
(1,8) ->    9    -7     7     8   1/8     8     8 (No)
(8/3,3) -> 17/3  -1/3   1/3     8   8/9   9/8     8 Yes!

We have a winner: 8 divided by 1/3 is 24.  Now, how did we get here?
Tracing back, we find that the corresponding expression is

8        8
------- = ----- = 8*3 = 24
3 - 8/3    1/3

So, this wasn't pretty, but I knew that I was covering all the bases,
and that I wasn't touching any of them more than once.  It's nice when
you can 'see' the answer directly, which sometimes happens; but when
that doesn't work, it's good to have a backup plan.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 08/06/2002 at 16:20:56
From: Roger Percival
Subject: Thanks

Thank you so much for the solution to my problem--it's great!

Warm regards,
Roger
```
Associated Topics:
Elementary Puzzles
Middle School Puzzles

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