Japanese Sums: "Two and One" by Thomas Snyder | Pencil Puzzle May 10, 2026

Added: 2026-05-16

[00:00:00]Hello, let's solve a Japanese sums puzzle. This is called two and one. It's by Thomas Snyder and I sourced this from GM Puzzles. There's a link in the description if you want to try it yourself.

[00:00:10]Let's go over the rules. So, what we're going to be doing is putting numbers in some cells.

[00:00:15]Like maybe a one here, two here, three here, four here, whatever. We're going to put some numbers in.

[00:00:20]And the ones cells that don't have numbers in them, we're going to shade.

[00:00:26]And so maybe like that. And so what we need to do is first of all, we cannot repeat any numbers in a row or a column.

[00:00:34]The numbers can only be one to six.

[00:00:36]And these clues outside the grid tell us the the sum of contiguous runs of numbers.

[00:00:44]So, for example, I could put one, two, three here and this is in order. So, I need a sum of one, then some shading between so some unfilled cells in between that are shaded, and then a two, and then more shading, and then the three.

[00:00:57]So, let's do a a more interesting like this six and seven, right? I could shade this one and this one.

[00:01:04]Then I could make these two add to six like one, five.

[00:01:08]I'd make these two add to seven like four, three.

[00:01:10]And so you can see that we have a shaded cell, and then we have a contiguous run of numbers that add to six, that matches that.

[00:01:18]Then another shaded cell, that's important, and then a contiguous run of numbers that add to seven.

[00:01:23]The question marks mean you just need a run there, but we're not giving you what the sum is.

[00:01:28]And I believe that should do it. There's a link in the description again if you want to try it yourself, and I'm going to get started right now.

[00:01:36]All right. Um first thing I see are these 21s. Now remember we can't repeat a digit in a column.

[00:01:41]And so we're going to have to use the entire column to get to 21 cuz one through six once each sums to 21. So, this entire the all three of these columns are um require a digit in them in every single cell.

[00:01:57]Um I don't think any of the other columns are useful at the moment, but these rows now are. So, a one is only going to be a one, and then we're going to have to shade after that.

[00:02:07]And the one's got to go here because we need a number in here. And so, this has to be match our one clue. This last one has to match our three clue. The one is gone, so we're not doing one two, so that's going to be the three. That's shaded, and then this has to match the two clue, which obviously can only be the two.

[00:02:22]We're not doing one plus one for many reasons.

[00:02:25]Cuz we'd repeat the one three times in this row then.

[00:02:29]Okay. This next one we're doing three four five.

[00:02:33]So, we're not connecting these two together because we're not going to be able to do three contiguous runs if we do this, right? Where would I shade to make three contiguous runs? So, this has to be shaded, and that makes this the five.

[00:02:46]Four five six has the same issue. Oops.

[00:02:49]So, we're going to shade there, and then this is our six.

[00:02:52]Six and seven does not have that issue.

[00:02:56]So, let's hold off on that for a second.

[00:02:59]This does need three clues in it. So, these aren't going to connect for that reason. So, this is also shaded, and that tells us where our two and one go.

[00:03:06]Two and one just like that, and these are going to have numbers in them.

[00:03:10]So, this has to be question mark. So, we don't know what this is.

[00:03:14]This is also question mark, so we don't know what it is, but this is seven.

[00:03:17]And seven you can't do in one cell. You got to use at least two cells, which then tells us we're using exactly two cells.

[00:03:24]Lots of ways to add to seven in two. One six, two five, three four.

[00:03:29]Um, but this this is going to have to be the two vertically. Cuz if it was the one, how are we doing a shaded cell and then a two after it, right? Doesn't make sense. There's not enough room for that.

[00:03:38]So, this is the two, and it is going to be surrounded by shaded cells. So, this is our question mark. These three at least are eight, but we can't do four cells to add to eight because one two three four already adds to 10.

[00:03:50]So, that's going to be shaded, and then these two add to seven without the one, so it's going to be two five or three four.

[00:03:58]Um I'm holding off on penciling for now, but we may come back and pencil. This just is the six.

[00:04:05]And then this adds to seven. We can't do seven in more than um three cells.

[00:04:10]And so these these are going to be a one four pair.

[00:04:16]This is part of our four, so it's either going to be a one three.

[00:04:22]So, we do need a one in one of these two.

[00:04:25]So, this could be one three.

[00:04:29]Or this would be one two.

[00:04:32]So, this is either two or three um am I penciling? I guess I'm penciling now. This is either two or three, and this is either three or four.

[00:04:42]Here, the four could be a four here, or we could do one three.

[00:04:49]Oh, we can't do one three, cuz that breaks both of these options. One and three use up the ones, we can't do one two, and it uses up the three, so we can't do the three. So, this is just a four. This is shaded.

[00:04:59]And then this is It doesn't mean this is the one, does it?

[00:05:06]Let's hold off on on assuming that.

[00:05:08]Okay.

[00:05:08]Sorry, I'm clearing my shading for no reason. Oh, vertically here we need a two. That's going to go here.

[00:05:13]And that is unshaded. So, this has to be shaded. So, that's our one. That's our two. That's our four. And this can't be a one. All right. And then this adds to five, so this is two three.

[00:05:27]Okay. This adds to seven, so that's two five.

[00:05:30]Let's uh let's make that green. All right, we have all our shading, so we should just be able to fill out the rest of these.

[00:05:36]This is one four, so the four tells us that's one, that's four.

[00:05:39]This adds to seven.

[00:05:42]We have five and six available.

[00:05:44]So, we'd either be 5 2 or we're not 3 4.

[00:05:52]This could be 5 2 or 6 1.

[00:05:57]Okay.

[00:05:59]How about Oh, here we know this is a three. Does that help? So, we need to add to eight. We already have one, so these two add to seven. We're not doing 1 6, we're not doing 3 4, so this has to be 2 5. This 5 tells us that's two, that's five. And so we can just finish these columns. This is a one, and this is a six, and there we go.

[00:06:17]All right.

[00:06:19]Well, that that was fun. I I've done Japanese sums before in Sudoku form, but not the the pure Japanese sums.

[00:06:25]Sometimes these make images, they don't always have to make images, though. But, there are kind of variants of this where they'll they'll draw you a pretty picture.

[00:06:34]Cool.

[00:06:35]Uh well, you know what? It kind of does look like a two and a one. Uh yeah, to an extent. This could be interpreted as a 2 1.

[00:06:43]Maybe.

[00:06:44]All right. Well, anyway, how'd you do?