The Number Plate Game

Figure 1

Most number plates in Indonesia display a four digit number as shown in Figure 1 where the number is 1234. A game I like to play as I drive around or am driven around is to try to make a digit equation from these four digits. My rules are that only the four operations of addition, subtraction, multiplication and division are allowed together with exponentiation and brackets. Concatenation of digits is not allowed. Division can be of two types:

• a digit divided into the next digit e.g. 2 | 4 = 2 which reads 2 divided into 4
• a digit divided by the next digit e.g. 4 / 2 = 2 which reads 4 divided by 2

For the 1234 in Figure 1, the digit equation is very straightforward:$$ \begin{align}  -1 + 2 &= -3 + 4 \\ 1 &=1 \end{align}$$Theoretically there are \(10^4 = 10000 \) possible numbers ranging from 0000 to 9999. Any numbers with at least two zeroes will form a digit equation. Take 1009 as an example:$$ \begin{align} 1 \times 0 &= 0 \times 9 \\ 0 &=0 \end{align} $$If the digit 1 appears at least twice in the first three digits, then a digit equation is always possible. Take 1814 as an example:$$\begin{align} 1^8 &= 1^4 \\ 1 &=1 \end{align}$$Two consecutive digits are often helpful because they can be collapsed to a 1. Take 6599 as an example:$$\begin{align} 6 - 5 &= 9 \, /  \,9 \\ 1 &=1 \end{align}$$One or more zeroes are also helpful because raising any digit to the zero power produces a 1. Take 9032 as an example:$$\begin{align} 9^0 &= 3-2 \\ 1 &=1 \end{align}$$Brackets also prove very useful as shown in the example of 4894 where we bracket the 8 and 9 and make the 8 negative:$$\begin{align} 4^{-8+9} &= 4 \\ 4^1 &=4 \\ 4&=4  \end{align}$$Of course, forming a digit equation can require a little trickery. Take 3649 as an example:$$\begin{align} 3^{ 6-4}&= 9 \\ 3^2 &=9 \\ 9 &= 9 \end{align}$$Sometimes a digit equation is not possible according to the rules imposed. Take 3637 as an example. I can't see how to make a digit equation out of those four digits.

On the other hand, it's often possible to form digit equations in several different ways. Take our original 1234 as an example where another representations is possible:$$\begin{align} 1^2 &= -3+4 \\ 1 &=1 \end{align}$$It would be an interesting exercise to work out digit equations for all the numbers between 0000 and 9999 and to make a note of those numbers where an equation is not possible. In fact I've began this process in an Airtable database (link) and it would appear that 0124 is the first four digit number to thwart the creation of a digit equation. I'll keep working on the database and make additions to this post when I discover numbers of interest. It will of course take some time to complete.

See post titled Forming Equations from the Digits of a Number from the 14th of March 2024.

ADDENDUM: July 10th 2024

A variation of this game would be to see if one can get the numbers to total zero. This is very similar to the game already described and every solution to that game becomes a solution to this game because \(a=b \implies a-b=0\). This new, let's call it "zero game", allows for additional numbers plates containing the digit zero to qualify. For example 2470:$$2470 \rightarrow 2 \times 4 \times 7 \times 0 = 0$$Under the old rules this doesn't qualify but here the product of the digits is zero so it does. Another example would be 6772:$$6772 \rightarrow 6 \times  (7 - 7) \times 2 = 0$$Under the old rules, a digit equation could not be formed from these digits but here it can.