Reasoning · Grade 5-1 Making a Target Number

Problem

Erase one plus sign to hit the target

The digits 1 to 9 stand in order joined by plus signs. In each of three lines exactly one plus is erased. Erasing pushes the two neighbouring digits into a two-digit number. Erase so the lines come to 90, 72 and 117.
Your answer
How to solve
Strategy Make a Systematic List — There are only eight plus signs, so the whole candidate set is small enough to write down completely. But listing eight new expressions and adding each one from scratch is slow, so first I answer an easier question — what does the expression add up to before anything is erased? — and then look at what one erasing does to that total. The jump turns out to follow a very tidy pattern, which lets me write all eight possible totals in one sweep and read the three answers straight off the list. A quick re-addition of each finished line is the check.
1STEP 1

Find what the expression is worth before any erasing

Before erasing the value is 45.

1+2+3+4+5+6+7+8+9 = 40 + 5 = 45
2STEP 2

See what erasing one plus sign does to the total

Erasing one plus adds nine times the left digit.

56 - (5+6) = 56 - 11 = 45 = 9 × 5
3STEP 3

Turn that into a rule and list all eight possibilities

So the totals run 54 up to 117, eight in all.

new total = 45 + 9 × a: 54, 63, 72, 81, 90, 99, 108, 117
4STEP 4

Read the three answers off the list

In the list 90, 72 and 117 come from joining 5, 3 and 8.

90 = 45 + 9 × 5, 72 = 45 + 9 × 3, 117 = 45 + 9 × 8
5STEP 5

Check each finished line by adding it out

Adding them out shows all three work.

1+2+3+4+56+7+8+9 = 90; 1+2+34+5+6+7+8+9 = 72; 1+2+3+4+5+6+7+89 = 117
Answer
between 5 and 6, between 3 and 4, between 8 and 9
45 + 9 × 5 = 90
Every answer has to be bigger than 45, since gluing two digits can only push the total up, and no bigger than 117, which is what erasing the very last plus gives. All three targets — 90, 72 and 117 — sit inside that window, so the puzzle is fair. Each target minus 45 gives 45, 27 and 72, and each of those is a multiple of 9 (namely 9 × 5, 9 × 3 and 9 × 8), exactly as the place-value rule demands; a target such as 80 would have failed that test and had no answer at all. Finally, the eight possible totals are all different, so each line has exactly one solution and nothing has been overlooked.
Takeaway

Gluing two digits together just promotes the left one to the tens place, so the total always jumps by exactly 9 times that digit — which turns a search into a single division!

  • Find what the expression is worth before any erasing
  • See what erasing one plus sign does to the total
  • Turn that into a rule and list all eight possibilities
  • Read the three answers off the list
  • Check each finished line by adding it out