6 replies to this topic

[rookie1ja]

No. 10 - Back to the Cryptograms and Algebra Puzzles
Replace the same characters by the same numerals so that the mathematical operations are correct.
(AA)^B = ABBA

This old topic is locked since solution is already provided in the Spoiler below.
Pls visit New Puzzles section to see always fresh brain teasers.

Spoiler for Solution

No. 10 - solution
11³ = 1331

[JASON4P8C10]

(AA)(multiplied by itself B times) = ABBA

A=2 B=1
2x2 to the 1st = 2x1x1x2=4
AxA to the B = AxBxBxA

A=2 B=2
2x2 to the 2nd =2x2x2x2=16
AxA to the B = AxBxBxA


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[mjoeck]

The ABBA doesn't imply multiplication as variables next to each other usually do. In these types of problems, each letter is assumed to be in position as if base ten numbers had been transposed with different letters of the alphabet. For example: 16 could be represented by AB, FD, UR, LH, etc. but not by AA. The puzzle is asking you which 2-digit multiple of 11, when raised to a single digit power, yields a 4-digit answer whose first and last digits are the same as those in the multiple of 11 and whose middle two digits are the power it was raised to. For example, raising 11 (represented by "AA") to the 2nd power (represented by "B") would give 121 (or "ABA"). We need a four digit answer though, so we need to either choose a higher multiple of 11, a higher power, or both. Happy solving!

[guderman]

this one is too easy...

11^3 = 11 * 11 * 11 = 11 * 121 = 1331

[BoilingOil]

this one is too easy...

11^3 = 11 * 11 * 11 = 11 * 121 = 1331

And then you beat me to it a second time! I'm impressed

[LordHawke]

I thought the same thing, and that is true, as long as ABBA=A*B*B*A. Then I realized ABBA is a 4-digit number w/ with the 2nd and 3rd being the same and the 1st and 4th being the same.

(AA)(multiplied by itself B times) = ABBA

A=2 B=1
2x2 to the 1st = 2x1x1x2=4
AxA to the B = AxBxBxA

A=2 B=2
2x2 to the 2nd =2x2x2x2=16
AxA to the B = AxBxBxA


THANK YOU THANK YOU
I ACCEPT DONATION

[James T]

No. 10 - Back to the Cryptograms and Algebra Puzzles
Replace the same characters by the same numerals so that the mathematical operations are correct.
(AA)^B = ABBA

Spoiler for different interpretation

Now let's say instead of the letters being the positions of digits in the number, let them to multiplied against themselves.

Thus we have (A*A)^B = A*A*B*B
taking the log base A² of both sides:

log_(A*A)(A*A)^B = log_(A*A)(A*A*B*B)
B = log_(A*A)(A*A) + log_(A*A)(B²)
B = 1 + 2log_A*A B

Which can only be true if B = 1 and A can be any number.

Edited by James T, 09 April 2009 - 01:08 AM.