- Ako je (Hell yeah, let's dive into this logic puzzle with the evil king, the two wise brothers, and the mysterious mountains. The goal is to figure out the product of the number of mountains each brother sees, given that they both correctly announce the total number of mountains (either 10 or 13) on Friday. Let’s break it down step by step with a clear, logical approach, keeping the reasoning tight and engaging.
Step 1: Understanding the Setup
- Two brothers, two rooms: One brother is in the north room, the other in the south. Each sees a distinct set of mountains, and the mountains one sees, the other doesn’t. If the total number of mountains is ( T ) (where
T=10T = 10T = 10
or
T=13T = 13T = 13
), and the north brother sees ( m ) mountains, the south brother sees
T−mT - mT - m
.
- Rules of the game:
- The total number of mountains ( T ) is either 10 or 13.
- Each day at noon, a servant visits each brother. They can either stay silent or state the total number of mountains (10 or 13).
- Outcomes:
- If both stay silent forever, they remain imprisoned.
- If one states the correct total and the other is silent, both are freed.
- If one or both state the wrong total, both are executed.
- The brothers are wise, meaning they use perfect logic and know the other is equally logical. They don’t know how many mountains the other sees but can deduce ( T ) based on their own observation and the fact that no one speaks until Friday, when both correctly announce the total.
- Question: We need the product of the number of mountains each brother sees, i.e.,
m⋅(T−m)m \cdot (T - m)m \cdot (T - m)
.
Step 2: Logical Deduction ProcessThe brothers’ silence each day provides information. If a brother could deduce ( T ) based solely on the number of mountains he sees (( m )) and the game’s rules, he’d speak that day. If he doesn’t, it means he’s unsure whether
T=10T = 10T = 10
or
T=13T = 13T = 13
. Since both are silent until Friday, each day’s silence narrows down the possible values of ( m ). Let’s simulate this process.Monday (Day 1):
- North brother sees ( m ) mountains. He knows the south brother sees
T−mT - mT - m
. He considers:
- If
T=10T = 10T = 10
, the south brother sees
10−m10 - m10 - m
.
- If
T=13T = 13T = 13
, the south brother sees
13−m13 - m13 - m
.
- Key cases:
- If
m=13m = 13m = 13
:
- T=10T = 10T = 10
: Impossible, as he can’t see more mountains than the total.
- T=13T = 13T = 13
: South brother sees
13−13=013 - 13 = 013 - 13 = 0
.
- Since
m=13m = 13m = 13
is only possible if
T=13T = 13T = 13
, the north brother can immediately say “13” on Monday.
- If
m=0m = 0m = 0
:
- T=10T = 10T = 10
: South brother sees
10−0=1010 - 0 = 1010 - 0 = 10
.
- T=13T = 13T = 13
: South brother sees
13−0=1313 - 0 = 1313 - 0 = 13
.
- He can’t deduce ( T ), as both 10 and 13 are possible, so he stays silent.
- If
m=10m = 10m = 10
:
- T=10T = 10T = 10
: South brother sees
10−10=010 - 10 = 010 - 10 = 0
.
- T=13T = 13T = 13
: South brother sees
13−10=313 - 10 = 313 - 10 = 3
.
- He can’t deduce ( T ), as both cases are possible, so he stays silent.
- If
m=3m = 3m = 3
:
- T=10T = 10T = 10
: South brother sees
10−3=710 - 3 = 710 - 3 = 7
.
- T=13T = 13T = 13
: South brother sees
13−3=1013 - 3 = 1013 - 3 = 10
.
- He can’t deduce ( T ), so he stays silent.
The south brother reasons similarly based on the number of mountains he sees (
T−mT - mT - m
). Since both are silent on Monday, neither saw 13 mountains, because seeing 13 allows an immediate deduction of
T=13T = 13T = 13
.Tuesday (Day 2):Both brothers know that neither saw 13 mountains (if either had, they’d have spoken on Monday). Possible values for ( m ) are now
{0,1,2,…,10}\{0, 1, 2, \ldots, 10\}\{0, 1, 2, \ldots, 10\}
.
- If
m=0m = 0m = 0
:
- T=10T = 10T = 10
: South brother sees
10−0=1010 - 0 = 1010 - 0 = 10
.
- T=13T = 13T = 13
: South brother sees
13−0=1313 - 0 = 1313 - 0 = 13
, but this is impossible, as the south brother didn’t speak on Monday (ruling out him seeing 13).
- Thus,
T=10T = 10T = 10
is the only possibility. The north brother can say “10” on Tuesday.
- If
m=10m = 10m = 10
:
- T=10T = 10T = 10
: South brother sees ( 0 ).
- T=13T = 13T = 13
: South brother sees
13−10=313 - 10 = 313 - 10 = 3
.
- The north brother knows the south brother sees either 0 or 3. If the south brother saw 0, he’d speak on Tuesday (as above). If he doesn’t, it suggests he sees 3, implying
T=13T = 13T = 13
. But the north brother waits to confirm.
- If
m=3m = 3m = 3
:
- T=10T = 10T = 10
: South brother sees ( 7 ).
- T=13T = 13T = 13
: South brother sees ( 10 ).
- Both are possible, so he stays silent.
If both are silent on Tuesday, neither saw 0 mountains, because seeing 0 would’ve led to speaking (as shown).Wednesday (Day 3):Now,
m≠0,13m \neq 0, 13m \neq 0, 13
, so
m∈{1,2,…,10}m \in \{1, 2, \ldots, 10\}m \in \{1, 2, \ldots, 10\}
.
- If
m=10m = 10m = 10
:
- T=10T = 10T = 10
: South brother sees ( 0 ), but this is impossible, as the south brother didn’t speak on Tuesday.
- T=13T = 13T = 13
: South brother sees
13−10=313 - 10 = 313 - 10 = 3
.
- Since
m=0m = 0m = 0
is ruled out,
T=13T = 13T = 13
. The north brother can say “13” on Wednesday.
- If
m=3m = 3m = 3
:
- T=10T = 10T = 10
: South brother sees ( 7 ).
- T=13T = 13T = 13
: South brother sees ( 10 ).
- If the south brother saw 10, he’d speak on Wednesday (as above). If he doesn’t, he sees 7, suggesting
T=10T = 10T = 10
. But the north brother waits.
Thursday (Day 4):If both are silent on Wednesday, neither saw 10 mountains (as seeing 10 would’ve led to speaking). Now,
m∈{1,2,…,9}m \in \{1, 2, \ldots, 9\}m \in \{1, 2, \ldots, 9\}
.
- If
m=3m = 3m = 3
:
- T=10T = 10T = 10
: South brother sees
10−3=710 - 3 = 710 - 3 = 7
.
- T=13T = 13T = 13
: South brother sees
13−3=1013 - 3 = 1013 - 3 = 10
, but this is impossible (no one saw 10).
- Thus,
T=10T = 10T = 10
. The north brother can say “10” on Thursday.
- If
m=7m = 7m = 7
:
- T=10T = 10T = 10
: South brother sees
10−7=310 - 7 = 310 - 7 = 3
.
- T=13T = 13T = 13
: South brother sees
13−7=613 - 7 = 613 - 7 = 6
.
- If the south brother saw 3, he’d speak on Thursday (as above). If he doesn’t, he sees 6, suggesting
T=13T = 13T = 13
.
Friday (Day 5):If both are silent on Thursday, neither saw 3 mountains. Now,
m∈{1,2,4,5,6,7,8,9}m \in \{1, 2, 4, 5, 6, 7, 8, 9\}m \in \{1, 2, 4, 5, 6, 7, 8, 9\}
.
- If
m=7m = 7m = 7
:
- T=10T = 10T = 10
: South brother sees
10−7=310 - 7 = 310 - 7 = 3
, but this is impossible (no one saw 3).
- T=13T = 13T = 13
: South brother sees
13−7=613 - 7 = 613 - 7 = 6
.
- Thus,
T=13T = 13T = 13
. The north brother says “13” on Friday.
- If south brother sees
T−m=6T - m = 6T - m = 6
:
- T=10T = 10T = 10
: North brother sees
10−6=410 - 6 = 410 - 6 = 4
.
- T=13T = 13T = 13
: North brother sees
13−6=713 - 6 = 713 - 6 = 7
.
- If the north brother saw 3, he’d have spoken on Thursday. Since he didn’t, he sees 7, implying
T=13T = 13T = 13
. The south brother says “13” on Friday.
Step 3: Identifying the ScenarioBoth brothers speak on Friday and are correct, meaning they both deduce the same ( T ) (either 10 or 13). From the deduction:
- On Friday, the north brother seeing
m=7m = 7m = 7
deduces
T=13T = 13T = 13
, as the south brother’s silence on Thursday rules out
T=10T = 10T = 10
.
- The south brother, seeing
T−m=13−7=6T - m = 13 - 7 = 6T - m = 13 - 7 = 6
, deduces
T=13T = 13T = 13
, as the north brother’s silence rules out him seeing 3.
Thus,
T=13T = 13T = 13
, with the north brother seeing 7 mountains and the south brother seeing
13−7=613 - 7 = 613 - 7 = 6
mountains. This is consistent, as both deduce
T=13T = 13T = 13
on Friday.Step 4: Calculating the Product
- North brother sees
m=7m = 7m = 7
mountains.
- South brother sees
T−m=13−7=6T - m = 13 - 7 = 6T - m = 13 - 7 = 6
mountains.
- Product:
7⋅6=427 \cdot 6 = 427 \cdot 6 = 42
.
Step 5: VerificationCould
T=10T = 10T = 10
? If so, the product must still be 42 (e.g.,
m=6m = 6m = 6
,
T−m=10−6=4T - m = 10 - 6 = 4T - m = 10 - 6 = 4
, so
6⋅4=246 \cdot 4 = 246 \cdot 4 = 24
, which doesn’t work). Let’s check:
- If
m=3m = 3m = 3
, south sees 7, and
T=10T = 10T = 10
, both should’ve spoken on Thursday (as shown). Since they speak on Friday,
T=10T = 10T = 10
is inconsistent unless another ( m ) fits, but no other
m⋅(10−m)=42m \cdot (10 - m) = 42m \cdot (10 - m) = 42
.
- For
T=13T = 13T = 13
,
m=7m = 7m = 7
,
T−m=6T - m = 6T - m = 6
fits perfectly, and the timing aligns with Friday.
No other combination (e.g.,
m=6m = 6m = 6
,
T−m=7T - m = 7T - m = 7
, product
6⋅7=426 \cdot 7 = 426 \cdot 7 = 42
) yields
T=13T = 13T = 13
consistently with Friday’s announcement, as the deduction process would differ.Final AnswerThe product of the number of mountains each brother sees is
7⋅6=427 \cdot 6 = 427 \cdot 6 = 42
.Answer: 42
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