32

Begin with a flagrantly erroneous summation and a woefully vacant substitution table.


           234
         +   5                   Digit        2    3    4    5    6    7    8
        -------           Substitute digit    _    _    _    _    _    _    _
          5678


How can the substitution table be filled out to correct this summation?

This is almost too easy if you just follow these guidelines.

  • Assign 7 unique substitute digits from 0 through 9 for digits 2 through 8 in the table (one digit per digit)

  • Replace digits in the summation by their substitutes in the table (no other kinds of edits, as the summation and table should be taken at face value)

  • All numbers and digits are decimal (no notation tricks are involved)

  • No leading zeros in the total or either summand

  • The summation has a unique solution

Added:   Regular pretty much forces the resultant summation. allows the guidelines to attain it.

humn
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  • Should all numbers be positive integers? – Techidiot Nov 01 '16 at 08:40
  • All positive integers, @Techidiot, only digits are being substituted (ohhh, guess you're used to someone sounding impolite when addressing you by name) – humn Nov 01 '16 at 08:41
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    Haha :D No problems ("Apparently, this user prefers to keep an air of mystery about them.") :p BTW, this looks like more of a lateral thinking than arithmetic :) – Techidiot Nov 01 '16 at 08:47
  • One final question, by no leading zeros I assume you mean, no leading zeros in the number being added or resulted right? – Techidiot Nov 01 '16 at 09:15
  • Right, user @Techidiot, having no leading zeros throughout is meant to force a unique result for the summation. Signed – user $@$humn – humn Nov 01 '16 at 09:17
  • @humn I need to know if I am on to something or if I am thinking way too much out of the box. Is it possible that which part of the graphic is the summation and which part is the substitution table is not what it might seem?(for example, there are 7 lines on the left AND the right of the words "substitute digit" etc,,,) – stack reader Nov 01 '16 at 10:33
  • More lateral than necessary, @stack reader, the summation and substitution table can be taken at face value. The secret is more closely related to assumptions about the guidelines. – humn Nov 01 '16 at 10:37
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    If Ryan27 has the right answer, then wow, that's clever. +1. – Deusovi Nov 01 '16 at 17:13
  • Neat question. However "Sum other numbers" is a brutal title, folks. I tried to change it to "Sum other numbers, using a digit substitution table" – smci Nov 02 '16 at 23:09

4 Answers4

35

Making an assumption:-

That if a substitute digit is itself in the lookup table, it will be replaced again. 

 Digit               2    3    4    5    6    7    8
 Substitute digit    3    4    9    1    7    8    0

The Summation becomes:

999 + 1 = 1000 because:
2->3->4->9,
3->4->9,
4->9,
5->1,
5->1,
6->7->8->0,
7->8->0,
8->0

Process:

As the question states, if you follow the guidelines, it should lead you towards the answer

First, as mentioned in the question, there is one possible summation. It must be 999 + 1 = 1000 as a 3 digit number plus a 1 digit number must equal a 4 digit number, and the first digit of the 4 digit number has to be the same as the 1 digit number.

Then, knowing that 6,7,8 must equal 0 we can first assign any one of those digits the substitute digit of zero, lets choose 8.

Since 0 is now used (and the question states the substitute digits must be unique) in order for 6 or 7 to equal 0, the only substitute digit we can assign is 8 (since 8 = 0).

This same logic is then applied to 2,3,4 since they all need to equal 9

Ryan27
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4

 My first guess is as we are adding a 1-digit number to a 3-digit number,
 and the result is a 4 digit number, the first digit of the result must me a 1.

 So 5 -> 1

 There is only one 3-digit number that becomes a 4-digit number when you add 1 to it.
 The number 999. So 234 stands for 999.

 So 2 -> 9
 And 3 -> 9
 And 4 -> 9

 And since the result is 1000, 5678 stands for 1000.

 So 6 -> 0
 And 7 -> 0
 And 8 -> 0

 So in the end, the table is:

 2    3    4    5    6    7    8
 9    9    9    1    0    0    0

 Which gives the summation:

   999
 +   1
 -----
  1000
nl-x
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  • You are not using unique digits in the subs table. Check other two answers -> With unique numbers and without them. There is nothing new here I guess. – Techidiot Nov 01 '16 at 15:09
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    +1 from me. This is the exact logic that I used when working through the puzzle. Just 1 lateral-thinking step from here and you arrive at the same answer I did. – Ryan27 Nov 01 '16 at 15:13
3

Only thing I could think of (lateral)(Before the edits to original question) ->

enter image description here

Techidiot
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2

Well...

 999
+  1
----
1000

No one can say that I substituted more than one digit for each of the initial digits. Therefore, each of the initial digits has a unique substitute.

Verence
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  • That way of thinking allows many possible answers though. – stack reader Nov 01 '16 at 11:29
  • @stackreader Does it? No leading zeros and unique substitution for 5. – Verence Nov 01 '16 at 11:29
  • Sorry then, I'm afraid I don't understand your explanation. Maybe it's just me. – stack reader Nov 01 '16 at 11:33
  • ^vote with a note: This solution highlights 3 "unique" meanings of "unique": i) single (not what is meant in the puzzle statement); ii) different from others (as is meant in the puzzle statement); iii) light-heartedly ironic (as is meant by "3 "unique" meanings" in this statement). By the way, the ^vote is also for a part of the puzzle solved correctly here. – humn Nov 01 '16 at 11:36