##### (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.

**Solution**

3 x 24 = 72

3 + 24 = 27

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

**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.

**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.

**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.

**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.

42^{2 }+ 53^{2}+ 68^{2} = 24^{2 }+ 35^{2}+ 86^{2}

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

42^{2 }+ 58^{2}+ 63^{2} = 24^{2 }+ 85^{2}+ 36^{2}

See if you can find the remaining four permutations.

**Solution**

43^{2 }+ 52^{2}+ 68^{2} = 34^{2 }+ 25^{2}+ 86^{2}

43^{2 }+ 58^{2}+ 62^{2} = 34^{2 }+ 85^{2}+ 26^{2}

48^{2 }+ 52^{2}+ 63^{2} = 84^{2 }+ 25^{2}+ 36^{2}

48^{2 }+ 53^{2}+ 62^{2} = 84^{2 }+ 35^{2}+ 26^{2}

*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.

13^{2 }+ 42^{2}+ 53^{2} + 57^{2 }+ 68^{2}+ 97^{2} = 31^{2 }+ 24^{2}+ 35^{2} + 57^{2 }+ 86^{2}+ 79^{2}

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

13^{3 }+ 42^{3}+ 53^{3} + 57^{3 }+ 68^{3}+ 97^{3} = 31^{3 }+ 24^{3}+ 35^{3} + 57^{3 }+ 86^{3}+ 79^{3}

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

**Solution**

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

12^{2 }+ 32^{2}+ 43^{2} + 56^{2 }+ 67^{2}+ 87^{2} = 21^{2 }+ 23^{2}+ 35^{2} + 65^{2 }+ 76^{2}+ 78^{2}

12^{3 }+ 32^{3}+ 43^{3} + 56^{3 }+ 67^{3}+ 87^{3} = 21^{3 }+ 23^{3}+ 35^{3} + 65^{3 }+ 76^{3}+ 78^{3}

*The Moscow Puzzles (Boris Kordemsky p169, 1978)*