Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 465. (September 2015)

K. 465. A treasure trunk has an electronic lock mechanism controlled by eight switches. Every switch has two settings: on or off. The lock opens if each switch is on. It is possible to change the setting of any switch to the opposite. However, the electronic sensors will detect which switch has been manipulated, and as a result, three other switches will be automatically changed, too. (These automatic changes will not generate further switches changing.) The table below shows which switch induces which further switches to change. (For simplicity, the switches are numbered.)

 Number of switch manipulated 1 2 3 4 5 6 7 8 Numbers of further switches changing automatically 2, 5, 7 1, 3, 8 5, 6, 7 1, 6, 8 2, 3, 6 2, 5, 8 1, 3, 4 1, 4, 7

$\displaystyle a)$ Initially, every switch is off, except for 6 and 7. The trunk can now be opened by manually changing the setting of two appropriate switches. Which two?

$\displaystyle b)$ Initially, every switch is off, except for 7. Is it possible to open the trunk now by manipulating the appropriate switches?

(6 pont)

Deadline expired on October 12, 2015.

Statistics:

 153 students sent a solution. 6 points: Barta Ákos, Bertók Zsanett, Csikós-Nagy Máté, Gárdonyi Csilla Dóra, Gilicze Márton, Háder Márk István, Kertész Ferenc, Kovács 576 Kristóf, Pálvölgyi Szilveszter, Peschka Viktor, Pinke Jakab Zoltán, Póta Balázs, Rubovszky Cecília , Sal Dávid, Szűcs Leó, Varga 294 Ákos, Vas Miklós, Vass Gábor Dávid, Veres Kata, Zentai Flóra, Zsótér Laura. 5 points: Antalffy-Zsíros Attila András, Bognár Ádám, Bővíz Anna Bella, Csáfordi József, Csóka Zoárd, Deák Viktória, Debreczeni Tibor, Dékány Barnabás, Dobák Dániel, Gréczi Gergely Ádám, Hegedűs András, Keltai Dóra, Kiss 468 Péter, Kovács 439 Boldizsár, Kovács Hanna, Köpenczei Csenge, Kulcsár Szabó András, Lantos Viktor, Máté 446 Dávid, Mészáros 916 Márton, Miskolczi Abigél, Mónos Péter, Nagy Csaba Jenő, Régely András, Simon Dóra, Szántó Julianna, Szöllősi Brigitta. 4 points: 13 students. 3 points: 24 students. 2 points: 45 students. 1 point: 13 students. 0 point: 10 students.

Problems in Mathematics of KöMaL, September 2015