• Every cell has a digit from 0 to 9 or one of the four operation symbols +, −, ×, and /, signifying addition, subtraction, multiplication, and division respectively.
  • No digit appears more than once in a row. No digit appears more than once in a column.
  • Reading across each row, with adjacent digits concatenated into a multidigit number, the arithmetic expression yields 24.
  • No + or 0 is superfluous to its arithmetic expression (e.g. at the beginning of a row). Similarly, no two operation symbols are next to one another in the same row.
  • There are bold lines outlining some areas that we'll call cages.
  • Each cage contains (possibly multiple copies of) + or ×, but does not contain both of those.
  • Each cage's digits (not concatenated but taken one at a time), combined via + or × (whichever appears in it), yields 24.
  • All the −, /, and 1 symbols are placed to get you started, as are two additional symbols.
  • $\begingroup$ "cage's digits combined via + and x yields 24" but from which direction to which direction? for example the middle cage is huge and confusing where to start and finish it. $\endgroup$
    – Oray
    Jan 22, 2023 at 9:12
  • 4
    $\begingroup$ @Oray + and × are commutative operations. $\endgroup$
    – msh210
    Jan 22, 2023 at 9:30
  • $\begingroup$ The minus in the 5th role: does it change the sign of the following number or is it just ignored? $\endgroup$
    – Tode
    Jan 23, 2023 at 9:59
  • $\begingroup$ @Tode it changes the sign of something. What it changes the sign of depends on the order of operations. $\endgroup$
    – msh210
    Jan 23, 2023 at 10:25

1 Answer 1


The filled grid:

enter image description here

For a start there is only room for two numbers in the middle left cage. The only way to get to 24 here is by multiplication. Because of given operators this can only be in one position. enter image description here

Now in row 5 the first multiplication leads to a negative number, to get to a positive value there needs to be a plus in this row. To compensate for the negative and get to 24 we need at least 2 digits after the plus fixing its position. enter image description here

If the top right cage would contain an x multiplication of the number in it would always lead to a number higher than 24. Therefore it can only contain + and -. With the digits already placed there is only one way they can go there. Then the first digit in the 3rd row has to be either 2 or 4 (all 1s are already given, already a 3 in the row and with the numbers present a number in the 50s or higher would give a row sum that is too large). Because of this the topleft cage has to be a multiplication cage with only one place for the x. enter image description here

The middle left cage only has two positions for numbers, so this one has to be a multiplication cage as well. Since 2, 3, 4 are already in the first column, some simple deductions can be made here. The same holds for the right middle cage. However here the digit in row three has to be 6 or 8 to assure division from 1.. by it gives a value in the 20s.
enter image description here

The digit in row 3 column 1 cannot be 3 since if it were three the completed expression for the row would either contain a double 6 or a double 8. This fixes the 3 at row 1 column 1. The sum of the digits in the middle cage cannot go over 24. Because of this the sum of the digits in the expression in row 3 has to stay low. The only way to achieve this from the remaining options here is as in the image below. enter image description here

Things start falling into place now. Both row 4 and 5 can only be completed in a single way. enter image description here

The second row can now only be completed in one way and after that because the sum of the remaining digits in the top right cage is known also the first row can only be completed in one way.
enter image description here

Because the 4th and the 7th column are very much restricted now, this leads to one single way to fill row 6. This leads to a set of options for each of the digits in row 7. I found the solution here by playing around with these options. There might be a nicer way here. enter image description here

  • $\begingroup$ Well done. I'd appreciate your feedback on the puzzle quality. (I guess in chat would be best.) $\endgroup$
    – msh210
    Jan 22, 2023 at 14:31

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