Surface of a number

 

To define the surface S of a number N we form two substrings of digits b and h with a single cut through N; we then compute the product b*h (base*height = S).
 
The surface of 8 is zero (as b or h are missing).
The surface of 10 is also zero (as b = 1 and h = 0, for instance).
The surface of 11 is 1 (as 1*1 = 1).
The surface of 39 is 27 (as 3*9 = 27).
2021 has two distinct surfaces, depending on where you cut:  the cut 202/1 will produce S = 202*1 = 202 and the cut 20/21 will produce S = 20*21 = 420 (no substring can start with a leading zero, except the substring 0 itself).
1234567890 has 9 distinct surfaces (9 possible cuts), etc.
 
The interesting part is that every number is the surface of at least two other numbers;
17 (a prime number) is one of the surfaces of 117 (1*17) and one of the surfaces of 171 (17*1);
2021 is one of the surfaces of 4347 (as 43*47 = 2021) but also one of the surfaces of 12021 and 20211 (as 2021 = 1*2021 = 2021*1).

 
Almost all numbers > 111 have more than one surface. Those surfaces can be ordered from the smallest to the largest; 112 has two surfaces for instance: 1*12 = 12 and 11*2 = 22. We call the smallest (12) the SurfMin and the largest (22) the SurfMax.

 
2021 has SurfMin = 202 (= 202*1) and SurfMax = 420 (= 20*21);
1234567890 has SurfMin = 0 (= 123456789*0) and SurfMax = 1111111020 (= 12345678*90) .

 
A difficult task is to find an integer that produces a given surface.
For instance, what is the smallest integer with SurfMin = 108? Here is our modus operandi:
 
First we decompose 108 into 2*2*3*3*3;
as we have only two factors (b and h) to find, we list all possible pairs such that b*h = 108:
2*54
3*36
4*27
6*18
9*12
But also, swapping b and h:
12*9
18*6
27*4
36*3
54*2

We now arrange from smallest to biggest the above integers:

129, 186, 254, 274, 336, 363, 427, 542, 618, 912.
Properly cut, those integers will all lead to 108.
But as we want 108 to be the smallest possible product b*h (or h*b), we must eliminate all the products that are < 108, as they would ruin the solution we are looking for:

129 has a SurfMin = 1*29 = 29, which doesn’t fit (as 29 < 108);
186 has a SurfMin of 86 (= 1*86), which doesn’t fit either (86 < 108);
254 has a SurfMin of 100 (= 25*4), which still doesn’t fit (100 < 108);
274 has a SurfMin of 108 (= 27*4), which is ok, we have a hit.
The answer to the question: “What is the smallest integer with SurfMin = 108” is 274.
 
Let’s try to find a few sequences (my apologizes for the inevitable typos hereunder):

(Many thanks to Hans Havermann and Maximilian Hasler for their corrections; Maximilian has noticed that the sequence B is already in the OEIS, here)

In our next post we will talk about the volume of a number  !-)