II/1. Denimo, da so bile Rokove kocke enotske. Kvader je imel tedaj prostornino 500 . Če je bil dolg $a$, širok $b$ in visok $c$ enot, je veljalo $a b c=500$. Njegova površina je bila $P=2(a b+b c+a c)$, vrednost izraza $P$ pa pove tudi, koliko mejnih ploskev kock je na površju kvadra. Poiskati moramo torej najmanjšo vrednost površine kvadra. Predpostavimo lahko, da je $a \geq b \geq c$. Ker je $500=2^{2} \cdot 5^{3}$, so delitelji števila 500 števila $1,2,4,5,10,20,25,50,100,125,250$ in 500 (glej preglednico). Razberemo lahko, da je na površju kvadra najmanj 400 mejnih ploskev kock, iz katerih je sestavljen.

| $a$ |  | $c$ | $2(a b+b c+a c)$ |
| ---: | ---: | :---: | :---: |
| 500 | 1 | 1 | 2002 |
| 250 | 2 | 1 | 1504 |
| 125 | 4 | 1 | 1258 |
| 125 | 2 | 2 | 1008 |
| 100 | 5 | 1 | 1210 |
| 50 | 10 | 1 | 1120 |
| 50 | 5 | 2 | 720 |
| 25 | 20 | 1 | 1090 |
| 25 | 10 | 2 | 640 |
| 25 | 5 | 4 | 490 |
| 20 | 5 | 5 | 450 |
| 10 | 10 | 5 | 400 |

Ugotovitev, da ima kvader z dolžinami robov $a, b$ in $c$ prostornino 500 (enot) ........ 1 točka  
Idea o deliteljih števila 500 in njegovi delitelji ............................................................................ Povezava števila mejnih ploskev (enotskih) kock s površino kvadra ........................ 1 točka  
Iskanje najmanjše vrednosti površine pri različnih vrednostih dolžin robov ............. 2 točki

![](https://cdn.mathpix.com/cropped/2024_06_07_5a4348e8f3c4edef190ag-07.jpg?height=52&width=1690&top_left_y=1887&top_left_x=186)

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Let's assume that Roko's cubes were unit cubes. The volume of the cuboid was then 500. If its length was $a$, width $b$, and height $c$ units, then $a b c=500$. Its surface area was $P=2(a b+b c+a c)$, and the value of the expression $P$ also indicates how many boundary faces of the unit cubes are on the surface of the cuboid. We need to find the smallest value of the surface area of the cuboid. We can assume that $a \geq b \geq c$. Since $500=2^{2} \cdot 5^{3}$, the divisors of 500 are the numbers $1,2,4,5,10,20,25,50,100,125,250$ and 500 (see the table). We can see that the cuboid has at least 400 boundary faces of the unit cubes it is composed of.

| $a$ |  | $c$ | $2(a b+b c+a c)$ |
| ---: | ---: | :---: | :---: |
| 500 | 1 | 1 | 2002 |
| 250 | 2 | 1 | 1504 |
| 125 | 4 | 1 | 1258 |
| 125 | 2 | 2 | 1008 |
| 100 | 5 | 1 | 1210 |
| 50 | 10 | 1 | 1120 |
| 50 | 5 | 2 | 720 |
| 25 | 20 | 1 | 1090 |
| 25 | 10 | 2 | 640 |
| 25 | 5 | 4 | 490 |
| 20 | 5 | 5 | 450 |
| 10 | 10 | 5 | 400 |

Determining that a cuboid with edge lengths $a, b$ and $c$ has a volume of 500 (units) ........ 1 point  
The idea of the divisors of the number 500 and its divisors ............................................................................ The connection between the number of boundary faces (unit) of the cubes and the surface area of the cuboid ........................ 1 point  
Finding the smallest value of the surface area for different edge lengths ............. 2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_5a4348e8f3c4edef190ag-07.jpg?height=52&width=1690&top_left_y=1887&top_left_x=186)
1. Rok used 500 equally large cubes to build a cuboid. What is the minimum number of faces of the cubes that made up the faces of the cuboid?