PSAMATHE: the relational synthesis test

An untimed, adaptive online test of relational synthesis, from the PSAMATHE research edition P01-E01 (28 October 2026), for ages 16 to 80. Every problem is a small network to build: five to seven lettered nodes, the links the board allows, and a few windows of evidence (how many direct neighbors a node has, how many neighbors two nodes share, how many links there are in all, whether a path joins two nodes). Its four families ask you to reconstruct the one network that fits every window, to repair a starting network with the fewest changes, to build a counterexample that fits every window and makes a stated claim false, and to build a network that stays connected whichever one of its links is lost (on the hardest problems, whichever one or two). You answer by building the network on a board, link by link. Every network a problem accepts scores, not only the one a solution shows: the keys list them all, from an exhaustive computer check of every network each board allows.

It opens with 4 practice problems, one per family, the examples shown on this page. Answer each and lock it: you see at once whether you were right, a network that is right and the reasoning. Practice is never scored. Then the scored problems come one at a time, each chosen from your answers so far: the families take turns, ten problems each, and in each the model picks one of the problems that tell most about your level, for 40 scored problems; from band 3 up, a band opens only once you have solved the one below it consistently. Build your network on the board, link by link, and lock it; a locked answer cannot be changed, and you cannot go back. If you cannot decide, you may leave a scored problem unanswered: it scores no point. Nothing tells you whether a scored answer is right, and your result appears the moment the last one is locked.

PSAMATHE's problems come in four bands, with model difficulties of IQ 70, 100, 130 and 160 (the planning locations of the author's model PS-M02). The bands grow in structure: entry problems have five nodes and six to nine available links, central ones five or six nodes and seven to ten links, advanced ones six nodes and ten to twelve, and stretch problems seven nodes and fourteen links, with at least four windows that each matter (three for robust construction), every accepted network connected and using six to ten links, repairs of at least four changes, and robust networks that must survive the loss of any one or two links. These are structural demands, not measured difficulties: the IQ values of the bands are the model's assumptions until real first attempts are analyzed.

From the problems you were given and your answers to them, by the provisional model PS-A01: the difficulties of the author's model PS-M02 (a Rasch model with an assumed difficulty for each band) and PRONOE's way of choosing problems. The model's estimate of your ability, with a standard normal prior, is shown as an IQ (mean 100, SD 15) with its 95% interval. A problem scores its point when the network you lock meets its task: every window holds and, by family, the claim is false, no repair with fewer changes exists, or every node stays connected after the losses. Two people with the same number of right answers can receive different problems and different results. PSAMATHE is a research edition: no human calibration has been done yet (N = 0), so a score is a provisional position on the model's scale, not a percentile of the population, and no percentile is reported. The author's package gives PSAMATHE as a fixed, balanced form; on this site, as for every adaptive test, each scored problem is chosen from your answers so far, and the score comes from the same model. In simulation, the estimates are within about 4.5 to 7 IQ points of the truth from about IQ 70 to 145; above that, the model's prior holds estimates back toward the middle of the scale, so a very high result is a cautious one. Forty right answers place the estimate at IQ 182, and the scale is reported up to IQ 180+; an estimate below IQ 70, the level of the easiest problems, is reported as below IQ 70. Beside the IQ, as THETIS does, your result gives the rule complexity: the band, from 1 to 4, of the problems you solve half the time. It describes what you solved, so it depends on no population norm.

A high score must rest on consistent success at the bands below it (the ladder, from 5 November 2026, as THETIS has above IQ 180). From band 3 up, PSAMATHE climbs one band at a time: no problem above the band being cleared is given until it is solved consistently, which takes at least three right answers at that band, from at least two families, with at least two in three of its problems right; very consistently, from 160 up, at least four right, with at least three in four of its problems right. So a score of 135 or more needs band 3 (problems of IQ 130) solved consistently, 160 or more band 4 (IQ 160) very consistently. Until then a score is held just below the next step, however well the answers above it went; the result and the certificate say so.

PSAMATHE is untimed. From the moment you begin, your attempt stays open for 90 days: work in as many sittings as you like, on any device, with your private link, and you come back to the problem you left. Each answer is locked on the server as you give it. An attempt still open after 90 days is scored with the answers locked by then, provided it has at least 40 scored answers; otherwise it ends without an IQ.

Yes, whenever you like once your previous attempt is finished, and every attempt is on scored problems you have never been shown. Every problem counts as seen the moment it reaches your screen, together with every problem of the same structure under other node letters, under this email or another registered with the same name and year of birth; no later attempt meets it again. Practice is the same four problems every time: the examples shown on this page. Every attempt has its own fee and its own certificate. Register every attempt with the same email. Only first attempts go into the research data on the test's norms.

The US$19.99 fee, paid through PayPal before the attempt opens, covers one attempt: the practice problems with their explanations, the 40 scored problems, your result with its PDF certificate, and a copy by email. Each retake has its own fee.

Yes, from 5 November 2026. IQ 147 or higher meets the criterion of the Nereid Society and IQ 160 that of the Mesopelagic Society (with a one-time membership fee). Just below the criteria, IQ 135 to 146 qualifies for associate membership of the Nereid Society and IQ 180+ for subscription to the Aphotic Society, each with a one-time admission fee. The criteria of the Aphotic, Hadal and Exocosm Societies lie above PSAMATHE's scale, which ends at IQ 180+. Your fee includes the review of one society application. Your best result counts, as for THETIS, and it counts as PSAMATHE's ladder holds it: a high score must rest on consistent success at the levels below it (see “How is my score worked out?”). PSAMATHE is a research edition, scored with provisional model norms until enough first attempts are in, and the author reviews every result before admission.

To build PSAMATHE's norms from real people. They are re-estimated from first attempts only, anchored on examinees whose IQ is known from a supervised test, such as a Wechsler or Stanford-Binet scale given by a psychologist, Raven's or Cattell's test given under supervision, or a high-IQ society's supervised admission test. Giving yours is optional, and it never changes your own result: it only helps place the problems on the IQ scale for everyone. Only attempts with a clean integrity record are used, and a new calibration is applied only when, checked on people it was not fitted on, it comes closer to their known IQs than the scale in force. If you allow it, the author may write to you to confirm the score.

The answer keys and the bank never leave the server, and your result never shows which scored problems you solved. Only the problem on screen ever reaches your browser: the next one is chosen on the server after you lock your answer, from the answers you gave, so there is no list of problems to look up in advance, and a retake is always on problems you have never been shown. Each problem is drawn on the screen with a faint watermark of your name and certificate ID. While a problem is on screen, the test page records leaving the test window, screenshot shortcuts (Print Screen, Windows+Shift+S, Command+Shift+3, 4 or 5), printing (Ctrl+P or Command+P), copy, right-click and drag attempts, and AI assistant add-ons that insert themselves into the page; the server keeps its own times (when each problem was shown and when its answer was locked) and notes right answers given moments after a long absence, and many right answers in a few seconds. As on THETIS, it checks the answer pattern against the difficulty of the problems you were given (a pattern unlikely for one person at your level is noted, as is a result at the top of the scale), and practice problems are recorded too. The record is sealed when you finish, and its fingerprint is printed on your certificate. The author's integrity report of each attempt lists all of it, problem by problem. None of this changes a score, and unusual results are reviewed, never judged automatically. An untimed test cannot rule out every form of help, which is why you declare that you will work alone.

Your private link (shown when your test is ready, and kept in this browser) reopens your test on any device, at the problem you have reached. If you lose it, use “Recover my test” with the PayPal Transaction ID from your receipt and the email you registered with. Every locked answer is on the server.

Yes: the four example problems above, one per family, with a network that is right and the worked solution of each. They are the practice problems that open every attempt, and they are never scored.