(i) Two Digits

If we multiply and also add the number 9 to itself we get the following results:

9 x 9 = 81 and 9 + 9 = 18. These two digit results are the digital reverse of each other.

Consider the two numbers 3 and 24. In a similar manner, show that these two numbers produce digitally reversed answers.

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Solution

3 x 24 = 72

3 + 24 = 27

Find another pair of numbers that produce digitally reversed answers containing two digits.

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Solution

The numbers 2 and 47 produce digitally reversed answers.

2 x 47 = 94

2 + 47 = 49

(ii) Three Digits

Consider the two numbers 2 and 497. In a similar manner, show that these two numbers produce digitally reversed answers containing three digits.

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Solution

If we multiply and also add the number 497 with the number 2 we get the following results:

497 x 2 = 994 and 497 + 2 = 499.

These three digit results are the digital reverse of each other.

Find another pair of numbers that produce digitally reversed answers containing three digits.

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Solution

The numbers 2 and 263 produce digitally reversed answers.

263 x 2 = 526

263 + 2 = 265

(iii) Product Digits

There are two digit pairs having the same product when both numbers are reversed.

Consider the following sets of numbers.

12 x 42 = 21 x 24                12 x 84 = 21 x 48                13 x 62 = 31 x 26                23 x 96 = 32 x 69

12 x 63 = 21 x 36                24 x 63 = 42 x 36                26 x 93 = 62 x 39                36 x 84 = 63 x 48

See if you can find another 5 pairs with the same property.

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Solution

46 x 96 = 64 x 69                14 x 82 = 41 x 28                34 x 86 = 43 x 68

23 x 64 = 32 x 46                13 x 93 = 31 x 39

The Moscow Puzzles (Boris Kordemsky p166, 1978)

(iv) Sum of Squared Pairs

We can write a set of squared pairs giving the same sum when their digits are reversed.

422 + 532+ 682 = 242 + 352+ 862

There are another 5 permutations of these same set of digits. One of these is shown below.

422 + 582+ 632 = 242 + 852+ 362

See if you can find the remaining four permutations.

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Solution

432 + 522+ 682 = 342 + 252+ 862

432 + 582+ 622 = 342 + 852+ 262

482 + 522+ 632 = 842 + 252+ 362

482 + 532+ 622 = 842 + 352+ 262

The Moscow Puzzles (Boris Kordemsky p168, 1978)

(v) Sum of Powers

Now here is a remarkable set of numbers. 13, 42, 53, 57, 68 & 97.

When each is raised to the power of 1, their sum is the same when their digits are reversed.

13 + 42 + 53 + 57 + 68 + 97 = 31 + 24 + 35 + 75 + 86 + 79

When each is raised to the power of 2, their sum is the same when their digits are reversed.

132 + 422+ 532 + 572 + 682+ 972 = 312 + 242+ 352 + 572 + 862+ 792

When each is raised to the power of 3, their sum is the same when their digits are reversed.

133 + 423+ 533 + 573 + 683+ 973 = 313 + 243+ 353 + 573 + 863+ 793

Show that the same is true for the following set of numbers. 12, 32, 43, 56, 67 & 87.

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Solution

12 + 32 + 43 + 56 + 67 + 87 = 21 + 23 + 34 + 65 + 76 + 78

122 + 322+ 432 + 562 + 672+ 872 = 212 + 232+ 352 + 652 + 762+ 782

123 + 323+ 433 + 563 + 673+ 873 = 213 + 233+ 353 + 653 + 763+ 783

The Moscow Puzzles (Boris Kordemsky p169, 1978)