Question

Think of this puzzle What do you need to find a chosen number from this square?

$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$10$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$
$20$	$21$	$22$	$23$	$24$	$25$	$26$	$27$	$28$	$29$
$30$	$31$	$32$	$33$	$34$	$35$	$36$	$37$	$38$	$39$
$6$	$7$	$8$	$9$	$44$	$45$	$46$	$47$	$48$	$49$
$16$	$17$	$18$	$19$	$54$	$55$	$56$	$57$	$58$	$59$
$26$	$27$	$28$	$29$	$64$	$65$	$66$	$67$	$68$	$69$
$36$	$37$	$38$	$39$	$74$	$75$	$76$	$77$	$78$	$79$
$46$	$47$	$48$	$49$	$84$	$85$	$86$	$87$	$88$	$89$
$56$	$57$	$58$	$59$	$94$	$95$	$96$	$97$	$98$	$99$

Four of the clues below are true but do nothing to help in finding the number.

Four of the clues are necessary for finding it.

Here are eight clues to use:
a. The number is greater than $9$ .
b. The number is not a multiple of $69$ .
c. The number is a multiple of $7$ .
d. The number is odd.
e. The number is not a multiple of $11$ .
f. The number is less than $200$ .
g. Its ones digit is larger than its tens digit.
h. Its tens digit is odd.

What is the number?
Can you sort out the four clues that help and the four clues that do not help in finding it?
First follow the clues and strike off the number which comes out from it.

Like - from the first clue we come to know that the number is not from $76$ to $9$ . (strike off numbers from $1$ to $9$ ).
After completing the puzzle, see which clue is important and which is not?

Accepted Answer

From the given puzzle, you need to find a chosen number from the square.

$0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$

$10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$

$20$ $21$ $22$ $23$ $24$ $25$ $26$ $27$ $28$ $29$

$30$ $31$ $32$ $33$ $34$ $35$ $36$ $37$ $38$ $39$

$40$ $41$ $42$ $43$ $44$ $45$ $46$ $47$ $48$ $49$

$50$ $51$ $52$ $53$ $54$ $55$ $56$ $57$ $58$ $59$

$60$ $61$ $62$ $63$ $64$ $65$ $66$ $67$ $68$ $69$

$70$ $71$ $72$ $73$ $74$ $75$ $76$ $77$ $78$ $79$

$80$ $81$ $82$ $83$ $84$ $85$ $86$ $87$ $88$ $89$

$90$ $91$ $92$ $93$ $94$ $95$ $96$ $97$ $98$ $99$

Given eight clues to use are:
a. The number is greater than $67$ .
b. The number is not a multiple of $10$ .
c. The number is a multiple of $7$ .
d. The number is odd.
e. The number is not a multiple of $11$ .
f. The number is less than $200$ .
g. Its ones digit is larger than its tens digit.
h. Its tens digit is odd.

Given that four of the above clues are only necessary for finding the number.

Deductive reasoning helps us to conclude that a particular statement is true, because it is a special case of a more general statement that is known to be true.

The answer is $75$ , because,

Statement 'a' does not help because by following the other clues you can tell that you need a two-digit number.

Statement 'b' does not help because the one digit has to be larger than the tens-digit and the only multiple of $7$ and $10$ is $79$ and $0$ is smaller than the $80$ .

Statement 'c' helps because being a multiple of $7$ cancels out a lot of numbers that could have been possibilities.

Statement 'd' helps because being an odd number it too cancels out a lot of other possibilities.

Statement 'e' does not help because the only multiple of $7$ and $11$ is $85$ and the ones digit has to be bigger than the tens digit.

Statement 'f' does not help, as the required number is two-digits.

Statement 'g' helps because by using it there will be few numbers left.

Statement 'h' helps by using it only $35$ remains.

Hence, the puzzle can be rewritten as:

$90$ $1$ $2$ $3$ $4$ $96$ $6$ $7$ $8$ $9$

$10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$

$20$ $21$ $22$ $23$ $24$ $25$ $26$ $27$ $28$ $29$

$30$ $31$ $32$ $33$ $34$ $35$ $36$ $37$ $38$ $39$

$40$ $41$ $42$ $43$ $44$ $45$ $46$ $47$ $48$ $49$

$50$ $51$ $52$ $53$ $54$ $55$ $56$ $57$ $58$ $59$

$60$ $61$ $62$ $63$ $64$ $65$ $66$ $67$ $68$ $69$

$70$ $71$ $72$ $73$ $74$ $75$ $76$ $77$ $78$ $79$

$80$ $81$ $82$ $83$ $84$ $85$ $86$ $87$ $88$ $89$

$90$ $91$ $92$ $93$ $94$ $95$ $96$ $97$ $98$ $99$

Statements 'c', 'd', 'g' and 'h' helps and they only are enough to get the number.

Important Points to Remember in Chapter -1 - Proofs in Mathematics from Telangana Board Mathematics Class 9 Solutions

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