Using Excel Solver to Find the Solution to a Geometry Puzzle

The Monday Puzzle in the Guardian by Alex Bellos had a good problem to illustrate the use of Excel Solver. The link to the article is: https://www.theguardian.com/science/2025/feb/17/did-you-solve-it-the-simple-geometry-problem-that-fools-almost-everyone.

Restated, the problem is to divide an 11 x 11 square into five rectangles so that the ten side lengths are the whole numbers from 1 to 10? 

Below is the 11 x 11 square with five rectangles and the ten sides labeled A through J. 

To use Excel Solver to find a solution, ten cells of the spreadsheet will be designated as the decision variables. In the attached spreadsheet, these are simply the cells A1 through J1. Solver will change their values until a solution is found. 

In rows 3 through 8 are 6 constraints:

A+D=11

C+F=11

E+H=11

G+B=11

H+J+D=11

B+I+F=11

Also in Solver, additional constraints are entered:

All values, A through J, are => 1 and <= 10.

All values, A through J, are integers.

All values, A through J, are different.

With these constraints entered, click the Solve button, and one of the solutions is displayed:

A=6, B=3, C=10, D=5, E=9, F=1, G=8, H=2, I=7, J=4

eleven by eleven square into 5 rectangles.xlsx

More about Solver: https://jamesmacmath.blogspot.com/2021/12/optimization-solutions-using-microsoft.html

Project Euler - Part 2

Five years ago I started the Project Euler Challenge. At the time, I was learning Python and used the challenge as a way to develop my programming skill. 

My prior post about time challenge: https://jamesmacmath.blogspot.com/2020/06/project-euler.html

Over the last two years, I've been learning Mathematica: On-line version. I decided to revisit my Project Euler project and found that I was able to complete several more of the problems using Mathematica. There are a total of 929 posted problems; I've only solved 50, so I have a long way to go.

A380198 Contribution to the OEIS

Difference between pi(2^n) and the integer nearest to 2^n / log(2^n).

0

-2, -1, 0, 0, 2, 3, 5, 8, 15, 24, 40, 72, 119, 212, 360, 633, 1128, 1989, 3580, 6386, 11537, 20897, 37980, 69354, 127336, 234054, 431877, 799754, 1484440, 2763961, 5156791, 9644970, 18080775, 33959344, 63902732, 120474951, 227515953, 430345298, 815241632

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..92

FORMULA

a(n) = - A053622(2^n).

a(n) = A007053(n) - A050499(2^n).

EXAMPLE

n 2^n pi(2^n) round(2^n/log(2^n) a(n)

------------------------------------------------

1 2 1 3 -2

2 4 2 3 -1

3 8 4 4 0

4 16 6 6 0

MATHEMATICA

Table[PrimePi[2^n]-Round[2^n/Log[2^n]], {n, 39}]

CROSSREFS

Cf. A000720, A007053, A050499, A053622, A057835.

Sequence in context: A378340 A063181 A247310 * A059220 A261630 A301503

Adjacent sequences: A380189 A380190 A380197 * A380208 A380209 A380210

KEYWORD

sign,new

AUTHOR

James C. McMahon, Jan 16 2025

STATUS

approved