PROBLEM MA 102 - Crux Mathematicorum Vol. 47 No 1

 En, pag4

Fr, pag 5

ANSWER CiP  $704$ squares

 

               Solution CiP

              Another grid of 31 matches has an area of 10 squares:

          A grid with $b$ matches horizontally and $a$ matches vertically contains $a \cdot (b+1)+b \cdot (a+1)$ matches. We have the equation

$2\cdot a \cdot b +a+b=337$

$\Leftrightarrow \;4ab+a+b=674$ $\;\Leftrightarrow \;(2a+1)\cdot (2b+1)=675$.  

 Examining all the decompositions into two factors of the number $675$ we obtain the table below.

$$\begin{matrix}  &2a+1  &3  &5  &9 &15   &25 \\  &2b+1  &225  &135  &75  &45  &27 \\  &a  &1  &2  &4  &7  &12 \\  &b  &112  &67  &37  &22  &13 \\  &area  &112  &134  &148  &154  &156 \end{matrix}$$

 The total area is $112+134+148+154+156=704.$

$\blacksquare$ 

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Tracking Number: 11633
Received:        Tue Feb  9 13:38:13 2021

From:  Petre Ciobanu
       Scoala Gimnaziala "Samuil Micu" SADU
       Sibiu, Romania
Email: ptr.ciobanu@gmail.com

Type:  Solve a MathemAttic Problem
       (problem MA102)

Files:
  MA102.pdf


Comments:
See my Blog
https://ogeometrie-cip.blogspot.com/2021/02/problem-ma-102-crux-mathematicorum-vol.html

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Fisierul Sursa Latex

https://artofproblemsolving.com/texer/rypsefyz

Crux Mathematicorum <crux@cms.math.ca>	15:39 (acum 1 minut)
către eu

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Added Jul 13, 2021

Good answer see V47n06, pag 278

     In solving them, they count in more detail the total number of matches:

"There are $(a+1)$ rows  of horizontal mathces, each containing $b$ matches. Similarly, there are $(b+1)$ columns of vertical matches, each containing $a$ matches. So the total number of matches is 

$(a+1)b+(b+1)a=$...

 =end added=