There is no solution if A to I each uniquely represent a digit 1 to 9, because the 9's remainder of ABC+DEF+GHI (0) cannot equal the 9's remainder of 2014 (=7).
Let me explain. The 9's remainder, or digital root (DR), is obtained by adding the digits of a number, adding the digits of the result, and so on till a single digit results. If the result is 9, the DR is zero and it's the result of dividing the number by 9 and noting the remainder only. E.g., 2014 has a DR of 7.
When an arithmetic operation is performed, the DR is preserved in the result. So the DRs of individual numbers in a sum give a result whose DR matches. If we add the numbers 1 to 9 we get 45 with a DR of zero, but 2014 has a DR of 7, so no arrangement can add up to 2014. If 2 is replaced by 0 in the set of available digits, the DR becomes 7 (sum of digits drops to 43, which has a DR of 7).
410+735+869=2014 is just one of many results of applying the following method.
Look at the number 2014 and consider its construction. The last digit is the result of adding C, F and I.
The result of addition can produce 4, 14 or 24, so a carryover may apply when we add the digits in the tens column, B, E and H. When these are added together, we may have a carryover into the hundreds of 0, 1 or 2. These alternative outcomes can be shown as a tree.
The tree:
04 >> 11, >> 19:
............21 >> 18;
14 >> 10, >> 19:
............20 >> 18;
24 >> 09, >> 19:
............19 >> 18.
The chevrons separate the units (left), tens (middle) and hundreds (right).
The carryover digit is the first digit of a pair. For example, 20 means that 2 is the carryover to the next column.
Each pair of digits in the units column is C+F+I; B+E+H in the tens; A+D+G in the hundreds.
Accompanying the tree is a table of possible digit summations appearing in the tree. Here's the table:
{04 (CFI): 013}
{09 (BEH): 018 036 045 135}
{10 (BEH): 019 037 046 136 145}
{11 (BEH): 038 047 137 056 146}
{14 (CFI): 059 149 068 158 167 347 086 176 356}
{18 (ADG): 189 369 459 378 468 567}
{19 (BEH/ADG): 379 469 478 568}
{20 (BEH): 389 479 569 578}
{21 (BEH): 489 579 678}
{24 (CFI): 789}
METHOD:
We use trial and error to find suitable digits. Start with units and sum of C, F and I, which can add up to 4, 14 or 24. The table says we can only use 0, 1 and 3 to make 4 with no carryover. The tree says if we go for 04, we must follow with a sum of 11 or 21 in the tens. The table gives all the combinations of digits that sum to 11 or 21.
If we go for 11 the tree says we need 19 next so that we get 20 with the carryover to give us the first two digits of 2014.
See how it works?
Now the fun bit. After picking 013 to start, scan 11 in the table for a trio that doesn't contain 0, 1 or 3. There isn't one, so try 21. We can pick any, because they're all suitable, so try 489. The tree says go for 18 next. Bingo! 567 is there and so we have all the digits: 013489576.
We have a result for CFIBEHADG=013489576, so ABCDEFGHI=540781693. There are 27 arrangements of these because we can rotate the units, tens and hundreds independently like the wheels of an arcade jackpot machine. For example: 541+783+690=2014. Every solution leads to 27 arrangements.
See how many you can find using the tree and table!
