Why Can We Make 10?

1. 10 really is easy to make

Let the four digits be (a, b, c, d).

The allowed operations are:

The digits may be rearranged freely, and parentheses may be placed in any valid way.

There are five binary-tree parenthesization structures for four operands. The three operators can be chosen in 4³ = 64 ways, and if all four digits are distinct, there are 24 permutations.

Therefore, the maximum number of raw candidates is:

Intermediate values were stored as reduced fractions rather than floating-point numbers, avoiding cases where something merely “close to 10” might accidentally be classified as exactly 10.

First, I examined all 9⁴ = 6561 ordered quadruples where each digit ranges from 1 through 9, and recorded which integers each quadruple could produce.

The easiest integer to make was not 10.

The most common target was 2, reachable by 6549 of the 6561 quadruples. It was followed by 3, 4, 6, 1, 5, and 7. 10 ranked eighth.

So 10 is not the easiest number to make.

But it is certainly an easy one.

2. Ignoring digit order

There is a problem with counting all 6561 ordered quadruples.

In the context of the 10 Puzzle, 1234 and 4321 are really the same problem.

So instead, consider only:

The number of such multisets is:

Of those 495 digit combinations, 447 can make 10.

That leaves only 48 combinations that cannot make 10.

Looking at those 48 revealed a remarkably clean property.

There are C(9,4) = 126 ways to choose four distinct digits from 1 through 9.

Every one of those 126 combinations can make 10.

Every combination that fails to make 10 contains at least one repeated digit.

This was one of the simplest and most beautiful properties found in the experiment.

3. Repetition reduces expressive freedom

Part of the reason is straightforward.

If all four digits are distinct, an abcd pattern has 24 permutations.

Repeated digits reduce that number:

The 48 combinations that cannot make 10 can be classified as:

• All distinct: 0
• One pair: 22
• Two pairs: 8
• Three identical digits: 14
• All identical: 4

Repeated digits reduce the structural freedom available to an expression, making holes in the set of reachable integers more likely.

That seems like a very natural explanation.

But this is where things begin to get strange.

4. Failure to make 10 does not mean low expressive power

Instead of looking only at 10, I counted how many different integers each digit combination could generate.

As expected, combinations with many repeated digits tended to generate fewer integers.

However, some combinations that cannot make 10, such as 6899, 5799, 5788, 3779, and 4899, can still generate a very large number of integers.

A combination that cannot make 10 is not necessarily bad at making numbers.

The interesting quantity therefore seemed to be not how many integers were reachable, but:

where the holes are

So I examined which integers from 1 through 24 were reachable for each of the 48 combinations that cannot make 10.

One particularly strange creature appeared.

6. Counting the number of paths

Next, instead of recording only whether an integer was reachable, I counted how many expressions in the search reached each integer.

Around 10, the counts for 2257 were:

10 appears zero times.

Interestingly, 14 appears only once.

What is that one expression?

First:

Then:

A human trying to make 14 from 2257 would probably start with 7 × 2 = 14.

But then the remaining 2 and 5 still have to be consumed.

The unique valid route approaches the problem almost backwards: make one-half from the other three digits, then divide 7 by it.

There is similarly only one route to −14:

So 2257 contains some extremely narrow paths.

There are two paths to 9.

One path to 14.

And still no path at all to 10.

7. Is the region around 10 simply sparse?

Looking only at 2257, this seems plausible.

There are only two paths to 9, and zero paths to 10.

Perhaps the region around 10 is already a low-density valley, and 10 simply happens to fall all the way to zero.

To test this idea, I compared other combinations that cannot make 10 while still reaching most integers from 1 through 24:

• 1399
• 3669
• 3779

This explanation also broke down.

For 3669:

There are dozens or even hundreds of paths immediately around 10.

Yet 10 itself remains at zero.

Looking at the missing integers from 1 through 24:

Each digit combination has holes in different places.

Since these combinations were selected specifically because they fail to make 10, the only hole they necessarily share is 10 itself.

A simple one-dimensional notion of “local sparsity” therefore did not seem sufficient.

8. Larger digits push results outward

Comparing against 1234 revealed another feature.

For 1234:

There are many paths to small integers.

By contrast, combinations such as 3779 show much lower frequencies within the range 1 through 24.

Perhaps larger input digits simply scatter results toward larger absolute values.

So for each digit combination, I calculated the frequency-weighted mean absolute value of the generated integers:

The results were:

3779 pushes its generated results outward more than ten times as strongly as 1234.

So the apparent sparsity of 3779 within 1 through 24 is at least partly an artifact of the observation window.

9. A surprisingly clean scale law

How, then, does the typical size of generated values depend on the input digits?

I first considered:

But this quantity increased as the digits became larger.

The scale of generated values therefore does not grow linearly with the sum of the input digits.

Using all 495 combinations, I performed a log-log regression of the form:

The result was:

The fit is remarkably strong.

For comparison, using the product of the four digits gives:

with:

The product is also highly predictive.

But surprisingly, the simple sum of the four input digits explains the typical scale of the generated values even better.

This was another unexpectedly clean empirical rule produced by the search.

10. And still, the hole at 10 remains

Could the combinations that fail to make 10 simply be combinations whose results spread farther outward than usual?

Using the regression above, I defined the normalized quantity:

Comparing the 447 combinations that can make 10 with the 48 that cannot:

On average, combinations that cannot make 10 do scatter their results about 8% farther outward.

But the difference is not decisive.

In particular, it does not explain something like 2257, which densely fills the central region while leaving almost exactly one hole at 10.

11. So what are the combinations that cannot make 10?

At first, I expected there to be a cleaner answer.

The number of even and odd digits.

The sum of the digits.

Their product.

The number of repeated digits.

The total number of reachable integers.

The spread of the generated values.

I expected some single boundary to exist, with combinations like 1177 and 2257 sitting naturally on one side of it.

At least within this experiment, no such simple property appeared.

Instead, three effects emerged.

The third effect is what remained hardest to reduce to a simple statistic.

For 2257, the hole is 10.

For 1399, the holes include 8, 10, and 23.

For 3669, they include 10, 19, and 23.

For 3779, they include 10, 16, and 21.

Which fractions can be produced along the way?

How can the four digits be divided into two subexpressions?

Can those subexpressions be connected to the target by one final addition, subtraction, multiplication, or division?

Small holes remain on the number line as a consequence of these discrete structural constraints.

10 is one of those holes.