Reading Comprehension Worksheets for Grade 6 - Free Printable

It looks like you've uploaded an image containing a math problem, but the text is not fully visible or readable in your message. However, based on what I can see, it appears to be a logic puzzle involving three logicians (A, B, and C) who are each given a positive integer, and they must deduce their own number based on information about the others' numbers and some logical reasoning.

Let me try to reconstruct the problem from the visible parts:

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Problem Summary (Reconstructed):

Three logicians — A, B, and C — are each given a positive integer written on their forehead. They can see the numbers on the other two people’s foreheads but not their own. The host tells them that:

> "One of your numbers is the sum of the other two."

Each logician knows this rule and that the others are also perfectly logical.

They are asked in turn whether they know their own number. The dialogue goes something like this:

- A says: "I don't know."
- B says: "I don't know."
- C says: "I don't know."
- Then A says: "Now I know my number!"

The question is: What are the possible numbers on their foreheads?

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Goal:

Determine the numbers on A, B, and C's foreheads, given that only A can deduce his number after hearing the others say they don’t know.

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Step-by-step Solution:

Let’s denote:
- $ a $ = number on A
- $ b $ = number on B
- $ c $ = number on C

We are told that one number is the sum of the other two, so one of these holds:
1. $ a = b + c $
2. $ b = a + c $
3. $ c = a + b $

All numbers are positive integers.

Logicians can see the other two numbers and use logic to deduce their own.

Let’s go through the statements.

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#### First Round:

A says: "I don't know."

A sees $ b $ and $ c $. He knows his number is either $ b + c $, or $ |b - c| $, because:
- If $ a = b + c $, then $ a $ is the sum.
- If $ b = a + c $, then $ a = b - c $
- If $ c = a + b $, then $ a = c - b $

But since all numbers are positive, $ a $ must be $ b + c $, or $ |b - c| $, but only if that result is positive.

So A considers:
- Is $ a = b + c $? → Possible.
- Is $ a = |b - c| $? → Only if $ b \ne c $, and the difference is positive.

But A doesn’t know which one it is. So he says “I don’t know” — meaning both possibilities are logically consistent with what he sees.

This implies that both $ b + c $ and $ |b - c| $ are valid candidates for $ a $, i.e., both could satisfy the “sum condition” depending on whose number is the sum.

But wait: the condition is that one number is the sum of the other two. So A cannot have more than one possibility unless both would satisfy the global constraint.

Let’s suppose A sees $ b $ and $ c $, and tries to figure out $ a $.

He considers:
- Case 1: $ a = b + c $
- Case 2: $ b = a + c $ → $ a = b - c $ (only if $ b > c $)
- Case 3: $ c = a + b $ → $ a = c - b $ (only if $ c > b $)

But only one of these can be true.

So A thinks: Could $ a = b + c $? Yes.

Could $ a = b - c $? Only if $ b > c $, and then $ b = a + c $, so $ b $ is the sum.

Could $ a = c - b $? Only if $ c > b $, and $ c = a + b $, so $ c $ is the sum.

So A has three possibilities, but only one can be correct.

But he says “I don’t know,” so he cannot eliminate any possibility.

That means: all possibilities are logically consistent with the rules — i.e., there is no way to tell which one is correct just from seeing $ b $ and $ c $.

So for A to not know, it must be that:
- $ b + c $ is a possible value for $ a $
- $ |b - c| $ is also a possible value for $ a $, and
- Both lead to valid configurations where one number is the sum of the other two.

But note: if $ b = c $, then $ |b - c| = 0 $, but numbers are positive integers, so $ a $ cannot be 0. So if $ b = c $, then $ a = b + c = 2b $ is the only possibility. So A would know his number.

Therefore, if $ b = c $, A would know $ a = 2b $. But A says “I don’t know,” so $ b \ne c $.

So $ b \ne c $, and both $ a = b + c $ and $ a = |b - c| $ are possible.

But now consider: if $ a = |b - c| $, then the larger of $ b $ or $ c $ is the sum.

For example, suppose $ b > c $. Then $ a = b - c $. Then $ b = a + c $, so $ b $ is the sum.

Similarly, if $ c > b $, $ a = c - b $, then $ c = a + b $, so $ c $ is the sum.

So for A to not know, both:
- $ a = b + c $ → $ a $ is the sum
- $ a = |b - c| $ → the larger of $ b $ or $ c $ is the sum

must be consistent.

So A sees $ b $ and $ c $, and cannot decide between:
- $ a = b + c $
- $ a = |b - c| $

So he says “I don’t know.”

Similarly for B and C.

Now B hears A say “I don’t know.” B sees $ a $ and $ c $. B also considers his own number $ b $.

B knows:
- $ b $ could be $ a + c $
- Or $ b = |a - c| $, if $ a \ne c $

But B also knows that A didn’t know. That gives B additional information.

Similarly for C.

Then finally, C says “I don’t know,” and then A says “Now I know.”

We need to find values such that this sequence of ignorance leads to A being able to deduce his number.

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Classic Version of This Puzzle:

This is a well-known logic puzzle. The typical solution involves small numbers.

Let’s suppose the numbers are small.

Try: $ a = 5 $, $ b = 3 $, $ c = 2 $

Check: $ 5 = 3 + 2 $ → yes, $ a $ is the sum.

Now simulate the conversation.

A sees $ b = 3 $, $ c = 2 $

A thinks:
- Could $ a = 3 + 2 = 5 $? Yes.
- Could $ a = |3 - 2| = 1 $? Then $ a = 1 $, and $ b = 3 $, $ c = 2 $, so $ b = a + c = 1 + 2 = 3 $ → valid.

So both $ a = 5 $ and $ a = 1 $ are possible.

So A says: “I don’t know.”

B sees $ a = 5 $, $ c = 2 $

B thinks:
- Could $ b = 5 + 2 = 7 $? Yes.
- Could $ b = |5 - 2| = 3 $? Then $ b = 3 $, $ a = 5 $, $ c = 2 $, and $ a = b + c = 3 + 2 = 5 $ → valid.

So both $ b = 7 $ and $ b = 3 $ are possible.

But B also knows that A said “I don’t know.”

Can B use that?

Yes.

Suppose $ b = 7 $. Then the numbers are $ a = 5 $, $ b = 7 $, $ c = 2 $. Then $ b = a + c = 5 + 2 = 7 $, so $ b $ is the sum.

Now, what would A see? A sees $ b = 7 $, $ c = 2 $. So A sees 7 and 2.

Then A would think:
- Could $ a = 7 + 2 = 9 $? Yes.
- Could $ a = |7 - 2| = 5 $? Yes, then $ a = 5 $, $ b = 7 $, $ c = 2 $, and $ b = a + c $ → valid.

So A would still have two options: 9 or 5. So A would say “I don’t know.”

So that’s consistent.

Now suppose $ b = 3 $. Then $ a = 5 $, $ b = 3 $, $ c = 2 $, and $ a = b + c $.

A sees $ b = 3 $, $ c = 2 $, same as before → A would still say “I don’t know.”

So both cases are consistent with A saying “I don’t know.”

So B cannot distinguish between $ b = 7 $ and $ b = 3 $. So B says “I don’t know.”

C sees $ a = 5 $, $ b = 3 $

C thinks:
- Could $ c = 5 + 3 = 8 $? Yes.
- Could $ c = |5 - 3| = 2 $? Yes, then $ c = 2 $, $ a = 5 $, $ b = 3 $, and $ a = b + c = 3 + 2 = 5 $ → valid.

So both $ c = 8 $ and $ c = 2 $ are possible.

Now, C knows A said “I don’t know,” and B said “I don’t know.”

Let’s test if both possibilities are consistent.

First, suppose $ c = 8 $. Then numbers are $ a = 5 $, $ b = 3 $, $ c = 8 $. Then $ c = a + b = 5 + 3 = 8 $, so $ c $ is the sum.

Now, what would A see? A sees $ b = 3 $, $ c = 8 $

A thinks:
- $ a = 3 + 8 = 11 $
- $ a = |3 - 8| = 5 $

Both are valid? Check:
- If $ a = 11 $, then $ a = b + c $ → valid.
- If $ a = 5 $, then $ c = a + b = 5 + 3 = 8 $ → valid.

So A would say “I don’t know.” OK.

Now B sees $ a = 5 $, $ c = 8 $

B thinks:
- $ b = 5 + 8 = 13 $
- $ b = |5 - 8| = 3 $

If $ b = 3 $, then $ c = a + b = 5 + 3 = 8 $ → valid.

If $ b = 13 $, then $ b = a + c = 5 + 8 = 13 $ → valid.

So B would also say “I don’t know.”

So both $ c = 8 $ and $ c = 2 $ are consistent with A and B saying “I don’t know.”

So C says “I don’t know.”

Now back to A.

A hears:
- B said “I don’t know”
- C said “I don’t know”

A originally saw $ b = 3 $, $ c = 2 $, and considered two possibilities:
1. $ a = 5 $ (so $ a = b + c $)
2. $ a = 1 $ (so $ b = a + c $, since $ 3 = 1 + 2 $)

Now A considers: if $ a = 1 $, would B and C have been able to say “I don’t know”?

Let’s test that.

Suppose $ a = 1 $, $ b = 3 $, $ c = 2 $

Then $ b = a + c = 1 + 2 = 3 $, so $ b $ is the sum.

Now, what would B see? B sees $ a = 1 $, $ c = 2 $

B thinks:
- $ b = 1 + 2 = 3 $
- $ b = |1 - 2| = 1 $

So $ b = 3 $ or $ b = 1 $

If $ b = 1 $, then $ c = a + b = 1 + 1 = 2 $ → valid.

If $ b = 3 $, then $ b = a + c = 1 + 2 = 3 $ → valid.

So B would say “I don’t know.”

Now C sees $ a = 1 $, $ b = 3 $

C thinks:
- $ c = 1 + 3 = 4 $
- $ c = |1 - 3| = 2 $

If $ c = 2 $, then $ b = a + c = 1 + 2 = 3 $ → valid.

If $ c = 4 $, then $ c = a + b = 1 + 3 = 4 $ → valid.

So C would also say “I don’t know.”

So both possibilities ($ a = 5 $ and $ a = 1 $) are consistent with the responses.

But now A says: “Now I know!”

Wait — but both possibilities are still consistent? So why does A now know?

Ah — but in the case $ a = 1 $, we have $ b = 3 $, $ c = 2 $

But let’s check what B would do if $ a = 1 $, $ b = 3 $, $ c = 2 $

B sees $ a = 1 $, $ c = 2 $

B considers $ b = 3 $ or $ b = 1 $

Now, B knows A said “I don’t know.”

What would A have seen if $ b = 1 $, $ c = 2 $? A would see $ b = 1 $, $ c = 2 $

Then A thinks:
- $ a = 1 + 2 = 3 $
- $ a = |1 - 2| = 1 $

So $ a = 3 $ or $ a = 1 $

But if $ a = 1 $, then $ b = 1 $, $ c = 2 $, and $ c = a + b = 1 + 1 = 2 $ → valid.

If $ a = 3 $, then $ a = b + c = 1 + 2 = 3 $ → valid.

So A would say “I don’t know.” OK.

But now, suppose $ b = 1 $, $ a = 1 $, $ c = 2 $

Then $ c = a + b = 1 + 1 = 2 $, so $ c $ is the sum.

Now, what would C see? C sees $ a = 1 $, $ b = 1 $

C thinks:
- $ c = 1 + 1 = 2 $
- $ c = |1 - 1| = 0 $ → invalid (since numbers are positive)

So only $ c = 2 $ is possible.

So C would know his number immediately.

But in our scenario, C said “I don’t know.”

So if $ b = 1 $, then C would see $ a = 1 $, $ b = 1 $, and would know $ c = 2 $, so would say “I know.”

But in reality, C said “I don’t know.”

Therefore, $ b $ cannot be 1.

So when B sees $ a = 1 $, $ c = 2 $, and considers $ b = 1 $, he realizes that if $ b = 1 $, then C would see $ a = 1 $, $ b = 1 $, and would know $ c = 2 $, so C would say “I know.”

But C said “I don’t know,” so $ b $ cannot be 1.

Therefore, B would deduce $ b = 3 $, and say “I know.”

But in our case, B said “I don’t know.”

Therefore, the assumption $ a = 1 $ leads to B being able to deduce his number, but B didn’t.

Hence, $ a \ne 1 $

Therefore, $ a = 5 $

So A can now deduce his number is 5.

Thus, the numbers are:
- A: 5
- B: 3
- C: 2

And the sum condition holds: $ 5 = 3 + 2 $

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Final Answer:

$$
\boxed{a = 5,\quad b = 3,\quad c = 2}
$$

Or any permutation where one number is the sum of the other two, and the logic chain works.

But in this case, the smallest such triple that satisfies the full conversation is $ \boxed{(5, 3, 2)} $, with A having 5.

Note: Other permutations may work, but the key is that the numbers are in a ratio such that the only consistent configuration after the series of "I don't know" statements is the one where the largest is the sum.

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Conclusion:

The numbers on the logicians’ foreheads are likely:

$$
\boxed{5,\ 3,\ 2}
$$

with A having 5, B having 3, and C having 2 (or similar assignment depending on who speaks).

But since the problem asks to solve the task, and based on the standard version, the answer is typically:

$$
\boxed{5,\ 3,\ 2}
$$