## Ken's Puzzle of the WeekDigits in the Blanks

________  +  ________  =  ________

Fill in the blanks with sequential digits to make the sum correct.  For example:  4 + 5 = 32.  Digits may be combined to make multi-digit numbers or used as exponents.

Source: Based on a puzzle in Barry R. Clarke's Brain Busters.
Solutions were received from Denis Borris, Claudio Baiocchi, Jacques Malan, Yaacov yoseph Weiss, Joseph DeVincentis, and Philippe Fondanaiche.  There are obviously many various solutions.  I've listed a few examples from all submissions are below.  I know there are more.  Feel free to add to these.

```1+2=3
3^2+5=14
76+5=3^4
178+56=234
381 + 546 = 927

Using the null-digit one can get "true" solutions as e.g.:
63+42=105
but also "cheated" ones. E.g., starting from the "natural" solution:
3^0+1=2
one can build other solutions by replacing the single digit 3 with 34, or
345, or 3456 and so on. Similar cheated examples come from:
3^4+1567^0=82
215+4^0=6^3
0^14567+8=2^3
0^35+16=2^4
0^145678+9=3^2;

There is a trivial solution for digits 1-9 in which all digits are in order.
1^(234567)+8=9

You may also accept
(01)^(234567)+8=9
if you allow leading zeros.

If digits do not need to be in order, any equation including 1 can be extended, for example:
(2^3)+(1^456780)=9

if 0 is included, everything can be added easily, for example:
4+5=3^2 --> (4^(16789^0))+5=(3^2)
which means any equation which fits the pattern can be extended.

Some "important" regular examples are:
342+756=1098 and 423+675=1098 (and relevant
permutations)

3^6 + 729 = 1458
3^6 + 549 = 1278
5^4 + 673 = 1298
6^4 + 283 = 1579

Using all 10 digits and no exponents, there are 72 solutions of the type
1978 + 56 = 2034 and 96 of the type 789 + 246 = 1035, counting a lot of trivial variations
like switching the addends or switching just the ones digits of the addends. That comes from the
alphametic solver here: http://www.tkcs-collins.com/truman/alphamet/alpha_solve.shtml

1 + 2 = 3
1^4 + 2 = 3
1^456789 + 2 = 3
173 + 286 = 459
4 + 2^5 = 36
5^2 + 3^4 = 106
4^3 + 56 = 120

```

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