The Four Digits Problem

W.W. Rouse Ball in his book Mathematical Recreations and Essays describes some variations on the old problem of representing numbers 1, 2, 3, … using four 4’s and various operations. He lists what he thinks are useful operations and whether they are considered “simpler” in some sense. In the table below, I have roughly followed his ordering, but have taken some liberties with some expressions and accepted an aesthetically more pleasing alternative. The main difference is that he places the denary scale (like 44) to be very simple, but I have downgraded that in order to prioritise the use of a 4 as the number four.

My order of importance is:

The additive operations: + and -
Multiply: ×
Divide: / or ÷
The denary notation, so 44 uses two digits 4.
A decimal point, so ·4 is allowed, even without the conventional zero before the decimal point. I have used a centred dot for this to make it more easily seen.
Recurring decimals, where the recurring sycle is shown by a bar above the cycling digits.
Square root symbol
Raising to an integer power, such as 4⁴ = 256.
Raising to a negative integer power, thus creating reciprocals,
Factorial function, n! = 1×2×3×...×n, so 4! = 24
Raising to a fraction as a power, which provides another way to take various roots
Taking a positive root other than a square root, so the degree is given in the formula
Taking a negative root, which is not a conventional way of taking reciprocals
Fractional root, which is highly irregular, but feasible with obvious meaning
Subfactorial function, written in a similar to a factorial except the ! symbol precedes the expression. By convention, postfix operators take precedence over prefix, so !n! means !(n!), but I have used brackets where necessary to ensure that the meaning is understood. The function is the number of ways of rearranging a set of n items such that none of them are in their natural position; it can be computed as !1 = 0; !n = !(n-1) + (-1)ⁿ . The first few values are !1 = 0, !2 = 1, !3 = 2, !4 = 9, !5 = 44, !6 = 265, !7 = 1854.

Rouse Ball also allows the use of a square root an infinite number of times, but that is a step I am unwilling to take, especially since as soon as we allow subfactorials we can achieve the equivalent.

The most common problem is to use four fours to make numbers in sequence 0, 1, 2, ... How far can we get? He claims to get to 877 with that set of allowable notations, and I agree. Here is a table of my proposals for fours 4’s.

Using each of the digits four times, we can reach to various lengths. Rouse Ball does not consider zero, and he limits himself and excludes the use of powers or subfactorials in his statements of how far we can get for each of the digits. The list below is the result of my further recent work and uses the full list of operations given above.

Four zeros gets us to 7.
Four ones gets us to 34.
Four twos gets us to 66.
Four threes gets us to 153.
Four fours gets us to 877, the largest number for four digits all the same.
Four fives gets us to 140.
Four sixes gets us to 46.
Four sevens gets us to 25.
Four eights gets us to 36.
Four nines gets us to 148.

If we consider what we can do with four different digits, then the obvious combination of using 1, 2, 3 and 4 was mentioned by Rouse Ball as well, but not using exactly the same set of rules. However, I looked at various sets of four numbers to see how far we can go with the rules given above.

The worst case is the set {0, 1, 6, 7}, which gets us only to 91.

The commonly considered case of the set {1, 2, 3, 4} makes numbers up to 652.

Many combinations can get over 1000, and the best case is the set {3, 4, 5, 7} which go as far as 1410.

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